\section{{\label{sec:intro}} Introduction}

Understanding of the steady state of non-equilibrium  systems is the subject of intense research. The typical situation is a solid in contact with two heat baths at different temperature. At the difference of equilibrium systems where the Boltzmann-Gibbs formalism provides an explicit description of the steady state, no equivalent theory is available for non-equilibrium stationary state (NESS).

In the last few years, efforts have been concentrated on stochastic lattice gases (\cite{S1}). For these latter precious informations on the steady state like the typical macroscopic profile of conserved quantities and the form of the Gaussian fluctuations around this profile have been obtained (\cite{S1}). Recently, Bertini, De Sole, Gabrielli, Jona-Lasinio and Landim proposed a definition of non-equilibrium thermodynamic functionals via a macroscopic fluctuation theory (MFT) which gives for large diffusive systems the probability of atypical profiles (\cite{BL1},\cite{BL2}) in NESS. The method relies on the theory of hydrodynamic limits and can be seen as an infinite-dimensional generalization of the Freidlin-Wentzel theory. The approach of Bertini et al. provides a variational principle from which one can write the equation of the time evolution of the typical profile responsible of a given fluctuation. The resolution of this variational problem is, however, in general very difficult and it has only been carried for two models : the Symmetric Simple Exclusion Process (SSEP) (\cite{BL2}) and the Kipnis Marchioro Presutti (KMP) model (\cite{BL3}). Hence, it is of extreme importance to identify simple models where one can test the validity of MFT.

The most studied stochastic lattice gas is the Simple Exclusion Process. Particles perform random walks on a lattice but jumps to occupied sites are suppressed. Hence the only interaction is due to the exclusion condition. The only conserved quantity by the bulk dynamics is the number of particles. In this situation, the heat reservoirs are replaced by particles reservoirs which fix the density at the boundaries. The KMP process is a Markov process composed of particles on a lattice. Each particle has an energy and a stochastic mechanism exchange energy between nearest-neigbor particles (\cite{KMP}).

The real motivation is to extend MFT for Hamiltonian systems (\cite{B-r}). Unfortunately, for these later, even the derivation of the typical profile of temperature adopted by the system in the steady state is out of range of the actual techniques (\cite{BLR}). The difficulty is to show that the systems behave ergodically, e.g. that the only time invariant measures locally absolutely continuous w.r.t. Lebesgue measure are, for infinitely extended spatial uniform systems, of the Gibbs type. For some stochastic lattice gases it can be proven but it remains a challenging problem for Hamiltonian dynamics.

We investigate here  the MFT for a system of harmonic oscillators perturbed by a conservative noise (\cite{Ber1}, \cite{BO},\cite{BBO}). These stochastic perturbations are here to reproduce (qualitatively) the effective (deterministic) randomness coming from the Hamiltonian dynamics (\cite{OVY}, \cite{LO}, \cite{FFL}). This hybrid system can be considered as a first modest step in the direction of purely Hamiltonian systems.

From a more technical point of view, SSEP and KMP are gradient systems and have only one conserved quantity. For gradient systems the microscopic current is a gradient (\cite{KL}) so that the macroscopic diffusive character of the system is trivial. Dealing with non-gradient models, we have to show that microscopically, the current is a gradient up to a small fluctuating term. The decomposition of the current in these two terms is known in the hydrodynamic limit literature as a \textit{fluctuation-dissipation equation} (\cite{EMY}).  In general, it is extremely difficult to solve such an equation.

Our model has two conserved quantities, energy and deformation, and is non-gradient. But fortunately, an exact fluctuation-dissipation equation can be established. In fact we are not able to apply MFT for the two conserved quantities but only for the temperature field which is a simple, but non-linear, functional of the energy and deformation fields.

The paper is organized as follows. In section \ref{SEC:2}, we define the model. In section \ref{SEC:3} we establish the fluctuation-dissipation equation and obtain hydrodynamic limits for the system in a diffusive scale. Section \ref{SEC:4} is devoted to a physical interpretation of the fluctuating term appearing in the fluctuation-dissipation equation. In section \ref{SEC:5} we compute the covariance of the fluctuation fields in the NESS by a dynamical approach and show the covariance for the energy presents a non-locality we retrieve in the large deviation functional (the quasi-potential).  The latter is studied in section \ref{SEC:6} for the temperature field.

\section{{\label{sec:2}} The Model}

We consider the dynamics of the open system of length $N$.  Atoms are labeled by $x \in \{1,\dots, N\}$. Atom $1$ and $N$ are in contact with two heat reservoirs at different
temperatures $T_{\ell}$ and $T_r$. Momenta of atoms are denoted by $p_1, \dots, p_{N}$ and the distance between particles are denoted by $r_1,
\dots, r_{N-1}$. The Hamiltonian of the system is given by
\begin{eqnarray*}
  {\ensuremath{\mathcal H}}^N = \sum_{x=1}^{N} {e}_x, \quad  {e}_x = \frac{ p_x^2 + r_x^2}2 \qquad x= 1,\dots, N-1\\
  {e}_{N} = \frac{p_{N}^2}2.
\end{eqnarray*}
We consider stochastic dynamics where the probability
density distribution on the phase space at time $t$, denoted
by $P(t, p, r)$, evolves following the Fokker-Planck equation
\begin{equation*}
\partial_t P = N^2 {\mathcal L}^* P
\end{equation*}
Here ${\mathcal L} = {\mathcal A} + \gamma {\mathcal S}+{\mathcal B}_{1,T_\ell} +{\mathcal B}_{N,T_r}$ is the generator of the process and ${\mathcal L}^*$ the
adjoint operator. The factor $N^2$ in front of ${\mathcal L}^*$ is here because we have speeded up the time by $N^2$, this corresponds to a diffusive scaling.

${\mathcal A}$ is the usual Hamiltonian vector field
\begin{eqnarray*}
{\mathcal A} = \sum_{x=1}^{N-1} (p_{x+1} - p_x) \partial_{r_x}
     + \sum_{x=2}^{N-1} (r_x - r_{x-1}) \partial_{p_x}\\
  + (r_1-\ell) \partial_{p_1} - (r_{N-1}-\ell) \partial_{p_{N}}
\end{eqnarray*}
The constant $\ell$ fix the deformation at the boundaries.

$\mathcal S$ is the generator of the stochastic perturbation and $\gamma>0$ is a positive parameter that regulates its strength. The operator $S$ acts only on momenta $\{p_x\}$ and generates a diffusion on the surface of constant kinetic energy. This is defined as follows. For every nearest neigbor atoms $x$ and $x+1$, consider the following one dimensional surface of constant kinetic energy $e$
\begin{equation*}
{\mathbb S}_e^1 =\{ (p_x,p_{x+1}) \in {\mathbb R}^2; p_x^2 +p_{x+1}^2 =e\}
\end{equation*}
The following vector field $X_{x,x+1}$ is tangent to ${\mathbb S}_{e}^1$
\begin{equation}
 \label{eq:4}
   X_{x, x +1} = p_{x+1} \partial_{p_x} -  p_x \partial_{p_{x+1}}
\end{equation}
so $X_{x,x+1}^2$ generates a diffusion on ${\mathbb S}_e^1$ (Brownian motion on the circle).  We define
\begin{equation*}
{\mathcal S}= \frac {1}{2} \sum_{x=1}^{N-1} X_{x, x +1}^2
\end{equation*}

${\mathcal B}_{1,T_\ell}$ and ${\mathcal B}_{N,T_r}$ are two boundary generators of Langevin baths at temperature $T_\ell$ and $T_r$
\begin{equation*}
{\mathcal B}_{x,T}= \frac 12 \left(T \partial_{p_x}^2 - p_x \partial_{p_x} \right)
\end{equation*}

The bulk dynamics conserve two quantities: the total energy ${\mathcal H}^N=\sum_{x=1}^{N} {e}_x$ and the total deformation ${\mathcal R}^N=\sum_{x=1}^{N-1} r_x$. The energy conservation law can be read locally as (\cite{Ber1}, \cite{BO})
\begin{equation*}
e_x (t) - e_x (0) = J^e_{x} (t) -J_{x+1}^e (t)
\end{equation*}
where $J^e_{x} (t)$ is the total energy current between $x-1$ and $x$ up to time $t$. This can be written as
\begin{equation*}
J^e_{x} (t)=N^2 \int_0^t j^e_{x} (s)ds + M_{x} (t)
\end{equation*}
In the above, $M_{x} (t)$ is a martingale, i.e. a stochastic noise with mean $0$. The instantaneous energy current $j^{e}_{x}$ can be written as
\begin{equation*}
j_{x}^e= -r_{x-1}p_{x} -\cfrac{\gamma}{2}\nabla(p_{x}^2)
\end{equation*}
The first term $-r_{x-1} p_x$ is the Hamiltonian contribution to the energy current while the noise contribution is given by the discrete gradient $-(\gamma/2) \nabla(p_{x}^2)=(\gamma/2)(p_{x}^2 -p_{x+1}^2)$.

Similarly, the deformation instantaneous current $j^{r}_{x}$ between $x-1$ and $x$ is given by
\begin{equation*}
j^{r}_{x}=-p_{x}
\end{equation*}

We denote by $\mu^{ss}=<\cdot>_{ss}$ the invariant probability measure for the process. In the case $T_\ell = T_r=T$, the system is in thermal equilibrium. There is no heat flux and the Gibbs invariant measure (or canonical measure) is a product Gaussian measure $\mu_{ss}=\mu^{T,\ell}$ depending on the temperature $T$ and the mean deformation $\ell$:
\begin{equation}
\label{eq:mu}
\mu^{T,\ell} = Z_{T}^{-1} \exp \left\{ -\cfrac{1}{2T}\sum_{x=1}^{N} p_x^2 -\cfrac{1}{2T}\sum_{x=1}^{N-1} (r_x-\ell)^2  \right\}
\end{equation}

\section{{\label{sec:3}}  Fluctuation-dissipation equation and  Hydrodynamic limit}
Diffusive interacting particle systems can be classified in two categories: gradient systems and non-gradient systems (\cite{KL}). For the first, we can write the curent of the conserved quantities as a spatial discrete gradient. For example SSEP and KMP process are gradient systems.  A powerful approach introduced by Varadhan (\cite{V}) to study non-gradient systems is to obtain a fluctuation-dissipation equation, meaning a decomposition of the current $j$ of conserved quantities as the sum of a microscopic gradient $\nabla h$ and of a fluctuating term of the form ${\mathcal L} u$:
\begin{equation}
\label{eq:fde0}
j=\nabla h + {\mathcal L}u
\end{equation}
where ${\mathcal L}$ is the generator of the interacting particle system. In fact, the equality (\ref{eq:fde0}) is only an approximation in a suitable Hilbert space (\cite{KL}).

Fortunately, for our system, we can write an equality like (\ref{eq:fde0}) without approximations.
The fluctuation-dissipation equation for the deformation current $j^r$ and the energy current $j^e$ is given by (\cite{Ber1})
\begin{equation}
\label{eq:fde}
\begin{cases}
j^{r}_{x}= -\gamma^{-1}\nabla(r_x) + {\mathcal L} h_x\\
j^e_{x}=\nabla\left[\phi_x \right]+{\mathcal L} g_x
\end{cases}
\end{equation}
where
$$\phi_x = \cfrac{1}{2\gamma}r_x^2 + \cfrac{\gamma}{2} p_x^2 +\cfrac{1}{2\gamma} p_x p_{x+1} +\cfrac{\gamma}{4}\nabla(p_{x+1}^2)$$
and
$$h_x= \gamma^{-1} p_x, \quad g_x =\cfrac{p_x^2}{4}+\cfrac{p_x}{2\gamma} (r_x + r_{x-1})$$

Assume that initially the system starts from a local equilibrium $<\cdot>$ with macroscopic deformation profile $u_0 (q)$ and energy profile ${\varepsilon}_0 (q)$, $q \in [0,1]$. This means that  if the macroscopic point $q \in [0,1]$ is related to the microscopic point $x$ by $q=x/N$ then at time $t=0$
\begin{equation*}
<r_{[Nq]} (0)> \to u_0(q), \quad <e_{[Nq]} (0)> \to {\varepsilon}_0 (q)
\end{equation*}
as $N$ goes to infinity. The currents are related to conserved quantities by the conservation law
\begin{eqnarray*}
\partial_t <r_{[Nq]}(t)> \approx -N \partial_q <j_{[Nq]}^r (t)>,\\
\partial_{t} <e_{[Nq]}(t)> \approx - N \partial_q <j_{[Nq]}^e (t)>.
\end{eqnarray*}
By (\ref{eq:fde}) and the fact that the terms  $N<{\mathcal L} h_x>$ and $N<{\mathcal L} g_x>$ are of order ${\mathcal O} (N^{-1})$ and do not contribute  to the limit (\cite{Ber1}) we get
\begin{equation*}
\begin{cases}
\partial_t <r_{[Nq]} (t)> \approx \gamma^{-1} \Delta <r_{[Nq]}(t)>\\
\partial_t <e_{[Nq]} (t)> \approx \Delta <\phi_{[Nq]} (t)>\\
\end{cases}
\end{equation*}
To close the hydrodynamic equations, one has to replace the term $<\phi_{[Nq]} (t)>$ by a function of the conserved quantities $<r_{[Nq]} (t)>$ and $<e_{[Nq]}(t)>$.

The replacement is obtained through a "thermal local equilibrium" statement (see \cite{Ber1}, \cite{BO}, \cite{ELS1},\cite{ELS2}, \cite{KL}, \cite{S1}) for a rigorous justification in the context of conservative interacting particle systems). We repeat here the arguments of \cite{BL3} for convenience of the reader. Thermal local equilibrium assumption corresponds to assume that each given macroscopic region of the system is in equilibrium, but different regions may be in different equilibrium states, corresponding to different values of the parameters. Let us consider an atom with position $q=x/N$ which is far from the boundary and introduce a very large number $2L+1$ of atoms in microscopic units ($L\gg 1$), but still an infinitesimal number at the macroscopic level ($(2L+1)/N \ll 1$). We choose hence $L=\epsilon N$ where $\epsilon \ll 1$ in order to have these two conditions. We consider the system in the box $\Lambda_L (x)$ composed of the atoms labeled by $x-L, \ldots, x+L$. The time evolution of the $2L+1$ atoms is essentially given by the bulk dynamics; since the variations of deformation and energy in the volume containing the $2L+1$ atoms changes only via boundary effects and we are looking at what happened after $N^2$ microscopic time units, the system composed of the $L$ atoms has relaxed to the micro-canonical state $\lambda_{{\bar r}_q (t), {\bar e}_q (t)}$ corresponding to the local empirical deformation ${\bar r}_q (t)$ and the local empirical energy ${\bar e}_q (t)$ in the box $\Lambda_L (x)$. This means that we can divide the observables into two classes, according to their relaxation times: the fast observables, which relax to equilibrium values on a time scale much shorter than $t$ and will not have any effect on the hydrodynamical scales and the slow observables which are locally conserved by the dynamics and need much longer times to relax. We can then replace the term $<\phi_{[Nq]} (t)>$ by $\lambda_{{\bar r}_q (t), {\bar e}_q (t)} (\phi_0)$. By equivalence of ensembles, in the thermodynamic limit $N \to \infty$ and then $\varepsilon \to 0$, this last quantity is equivalent to
\begin{equation*}
\cfrac{\gamma+\gamma^{-1}}{2} <e_{[Nq]} (t)> + \cfrac{\gamma^{-1}-\gamma}{4} (<r_{[Nq]} (t)>)^2
\end{equation*}
We have obtained the time evolution of the deformation/energy profiles $u(t,q)=\lim <r_{[Nq]} (t)>$, $\varepsilon (t,q)= \lim <e_{[Nq]} (t)>$ in the bulk. At the boundaries, Langevin baths fix temperature at $T_\ell$ and $T_r$. Hence it is more natural to introduce the couple of deformation/temperature profiles rather than deformation/energy profiles. The temperature profile $T(t,q)$ is related to $u(t,q)$ and $\varepsilon (t,q)$ by $\varepsilon (t,q) = T(t,q) +u(t,q)^2/2$. Deformation and temperature profiles evolve according to the following equations
\begin{equation}
\label{eq:hl}
\begin{cases}
\partial_t T = \cfrac{1}{2}(\gamma+\gamma^{-1}) \Delta T + \gamma^{-1}(\nabla u)^2,\\
\partial_t u =\gamma^{-1} \Delta u,\\
T(t,0)=T_\ell,\;\; T(t,1)=T_r,\\
u(t,0)=u(t,1)=\ell,\\
T(0,q)=T_0 (q),\; u(0,q)=u_{0}(q).
\end{cases}
\end{equation}

As $t$ goes to infinity, the system reaches its steady state characterized in the thermodynamic limit by a linear temperature profile ${\bar T}(q)=T_\ell + (T_r -T_\ell) q$ and a constant deformation profile ${\bar r}(q)= \ell$. The system satisfies Fourier's law and the conductivity is given by $(\gamma+\gamma^{-1})/2$ (\cite{BO}).

\section{{\label{sec:4}} Interpretation of the fluctuation-dissipation equation}

We have seen that functions $h_x$ and $g_x$ had no influence on the form of the hydrodynamic equations. This is well understood by the fact that they are related to \textit{first order} corrections to local equilibrium as we explain below.

Assume $T_{\ell (r)}=T \pm \delta T /2$ with $\delta T$ small. For $\delta T = 0$, the stationary state $<\cdot>_{ss}$ equals the Gibbs measure $\mu_{T}^{\ell}$ (see \ref{eq:mu}). If $\delta T$ is small, it is suggestive to try an ansatz for $<\cdot>_{ss}$ in the form:
$$\tilde{\mu}  = Z^{-1} \,\prod_x dp_x dr_x \exp\left(- \cfrac{1}{2T(x/N)} (p_x^2 + (r_x -\ell)^2) \right)$$
where $T(\cdot)$ is the linear interpolation on $[0,1]$ between $T_\ell$ and $T_r$. ${\tilde \mu}$ is the "local equilibrium" approximation of $<\cdot>_{ss}$. Let $f_{ss}$ be the density of the stationary state $<\cdot>_{ss}$ with respect to ${\tilde \mu}$, i.e. the solution of ${\mathcal L}^{*,T(\cdot)} f_{ss} =0$. Here ${\mathcal L}^{*,T(\cdot)}$ is the adjoint operator of ${\mathcal L}$ in ${\mathbb L}^2 ({\tilde \mu})$. It turns out that
\begin{eqnarray*}
{\mathcal L}^{*,T(\cdot)}&=& -{\mathcal A} +\gamma {\mathcal S} + B_{1,T_\ell} +B_{N,T_r} \\
&+&\cfrac{\delta T}{T^2}   \left( \cfrac{1}{N} \sum_{x=1}^{N-2} {\tilde j}^{e}_{x,x+1} -\ell \, \cfrac{1}{N} \sum_{x=1}^{N-1} {\tilde j}^r_{x,x+1}\right)\\
&+&\cfrac{\delta T}{T^2} \left( \cfrac{1}{N} \sum_{x=1}^{N-1} p_x p_{x+1} X_{x,x+1}\right) +\cfrac{\delta T}{4} (\partial^2_{p_1} -\partial^{2}_{p_N})\\
&+& {\mathcal O} ((\delta T)^2) +{\mathcal O}(N^{-1})
\end{eqnarray*}
where ${\hat j}^{e}$ and ${\hat j}^r$ are the energy and deformation currents for the reversed dynamics at equilibrium. They are obtained from $j^e$ and $j^r$ by reversing momenta $p \to -p$. Expanding $f_{ss}$ at first order $f_{ss}= 1 +\delta T\, v + o(\delta T)$, we get that for large $N$ and small gradient temperature $\delta T$, $v$ has to satisfy the following Poisson equation:
\begin{equation*}
(-{\mathcal A} + \gamma {\mathcal S}) v = T^{-2} \left(\cfrac{1}{N} \sum_{x=1}^{N-2} {\tilde j}^{e}_{x} -\ell \, \cfrac{1}{N} \sum_{x=1}^{N-1} {\tilde j}^r_{x}\right)
\end{equation*}
Let ${\hat v}$ the function obtained from $v$ by reversing momenta. By the fluctuation-dissipation equation (\ref{eq:fde}) we get
\begin{equation*}
{\hat v}=\cfrac{1}{NT^2}\sum_{x=1}^{N-1} (g_x- \ell h_x ) + {\mathcal O} (N^{-1})
\end{equation*}
Therefore the functions $g_x$ and $h_x$ are directly related to first order corrections to local equilibrium.

\section{{\label{sec:5}} Non-equilibrium Fluctuations and steady State Correlations}

Assume that initially the system starts from a local equilibrium $<\cdot>$ with macroscopic deformation profile $u_0 (q)$ and temperature profile $T_0 (q)$, $q \in [0,1]$. The time-dependant deformation fluctuation field $R_t^N$ and energy fluctuation field $Y_t^N$ are defined by
$$R_t^N (H)=\cfrac{1}{\sqrt N}\sum_{x=1}^{N} H\left (x/N \right)\left(r_x (t) - u\left(t,x/N\right) \right)$$
$$Y_t^N (G)=\cfrac{1}{\sqrt N}\sum_{x=1}^{N} G\left(x/N\right)\left(e_x (t) -\varepsilon(t,x/N) \right)$$
where $H,G$ are smooth test functions, $(T(t,\cdot), u(t,\cdot))$ are solutions of the hydrodynamic equations (\ref{eq:hl}) with $\varepsilon=T+u^2 /2$.

The fluctuation-dissipation equations (section \ref{SEC:3}) give (\cite{FNO}, \cite{S1}):
\begin{equation*}
\begin{cases}
R_t^N (H)=R_0^N (H)+ \cfrac{1}{\gamma}\int_{0}^t R_s^N(\Delta  H)ds+{\ensuremath{\mathcal M}}_t^{1,N}\\
Y_t^N (G)=Y_0^N (G)+\gamma \int_{0}^{t} Y_s^N (\Delta G)ds\\
\phantom{Y_t^N (G)} + \int_{0}^{t}ds\left\{\cfrac{1}{\sqrt{N}}\sum_{x\in {\mathbb T}_N}(\Delta G)(x/N)f_x(\omega_s)\right\}\\
\phantom{Y_t^N (G)} +{\ensuremath{\mathcal M}}_t^{2,N}
\end{cases}
\end{equation*}
where ${\ensuremath{\mathcal M}}^{1,N}$ and ${\ensuremath{\mathcal M}}^{2,N}$ are martingales and $f_x$ is the function defined by
$$f_x (\omega)= \cfrac{\left(\gamma^{-1}-\gamma\right)}{2} r_x^2-\left(\cfrac{1}{2\gamma}p_{x+1}p_x -\cfrac{\gamma}{4}\nabla^{*} p_x^2\right)$$

Covariance of the limit martingales are computed using standard stochastic calculus and thermal equilibrium property (\cite{S1}, \cite{FNO}):
\begin{equation*}
 \left<\left({\ensuremath{\mathcal M}}^{1,N}_t\right)^2\right> \rightarrow \cfrac{2}{\gamma}\int_0^t ds \int_{[0,1]}dq T(q,s)(\nabla H)^{2}(q)
\end{equation*}

\begin{equation*}
\begin{split}
 \left<\left({\ensuremath{\mathcal M}}^{2,N}_t\right)^2\right> \rightarrow \cfrac{2}{\gamma}\int_{[0,1]}dq \int_0^t ds u^{2} (q,s) T(q,s)(\nabla G)^{2}(q)\\
+(\gamma+\gamma^{-1})\int_0^t ds \int_{[0,1]}dq T^{2}(q,s)(\nabla G)^{2}(q)
\end{split}
\end{equation*}

\begin{eqnarray*}
 \left<{\ensuremath{\mathcal M}}^{1,N}_t {\ensuremath{\mathcal M}}^{2,N}_t\right> \rightarrow \\
\cfrac{2}{\gamma}\int_0^t ds\int_{[0,1]}dq u(s,q) T(s,q)(\nabla G)(q)(\nabla H)(q)
\end{eqnarray*}

Hence $R_t^N$ converges as $N$ goes to infinity to the solution of the linear stochastic differential equation:
\begin{equation}
\label{eq:R}
\partial_t R =\cfrac{1}{\gamma}\Delta R -\nabla\left[\sqrt{\cfrac{2}{\gamma}T(t,q)}W_{1}(t,q)\right]
\end{equation}
where $W_1(t,q)$ is a standard space time white noise.\\

The description of the limit for the energy fluctuation field is more demanding. We have first to close the equation. In order to do it, we use a "dynamical Boltzmann-Gibbs lemma" (\cite{KL}, \cite{S1}). Observables are divided into two classes: non-hydrodynamical and hydrodynamical. The first one are non conserved quantities and fluctuate in a much faster scale than the others (in the time scale where these last change). Hence, they should average out and only their projection on the hydrodynamical variables should persist in the limit. One expects there exist constants $C, D$ such that
\begin{eqnarray*}
\cfrac{1}{\sqrt N} \int_0^t ds \sum_{x=1}^{N} (\Delta G)(x/N) \left\{ f_x (\omega_s)\right.\\
\left. -C(r_x -u(s,x/N)) -D\left(\ensuremath{\mathcal E}_x-\varepsilon (s,x/N)\right)\right\}
\end{eqnarray*}
vanishes as $N$ goes to infinity. Constants $C$ and $D$ depend on the macroscopic point $q=x/N$ and on the time $t$. In order to compute these constants, we assume thermal local equilibrium. Around the macroscopic point $q$, the system is considered in equilibrium with a fixed value of the deformation $u(t,q)$ and of the temperature $T(t,q)$. The constant $C,D$ are then computed by projecting the function $f_x$ on the deformation and energy fields (\cite{KL}, \cite{S1}). If $\mu^{T,\ell}$ is the Gibbs equilibrium measure with temperature $T$ and mean deformation $\ell$ (the mean energy is then ${\varepsilon}={\ell}^{2}/2 +T$), we have $\Phi(\ell,{\varepsilon})=\mu^{T,\ell}(f_{x})=\varepsilon +{\ell}^{2}/2$ and then
\begin{equation*}
C=\partial_\ell \Phi (u(s,q), {\varepsilon} (s,q)), \quad D= \partial_{\varepsilon} \Phi (u(s,q),{\varepsilon} (s,q))
\end{equation*}

Therefore the time-dependant energy fluctuation field $Y_t^N$ converges as $N$ goes to infinity to the solution of the linear stochastic differential equation:
\begin{widetext}
\begin{equation}
\label{eq:Y}
\partial_t Y =\cfrac{1}{2}\left(\gamma+\cfrac{1}{\gamma}\right)\Delta Y +\cfrac{1}{2}\left(\cfrac{1}{\gamma}-\gamma\right)\Delta (u(t,q) R)
-\nabla\left[\sqrt{\gamma +\gamma^{-1}}T(t,q)W_{2}(t,q)+u(t,q)\sqrt{\cfrac{2T(t,q)}{\gamma}}W_1(t,q)\right]
\end{equation}
\end{widetext}
where $W_2 (t,q)$ is a standard space-time white noise independent of $W_1 (t,q)$.

Remark that the deterministic terms in (\ref{eq:R}) and (\ref{eq:Y}) result from linearizing the nonlinear equation as (\ref{eq:hl}).

We now compute the fluctuations fields for the NESS $<\cdot>_{ss}$ which is obtained as the stationary solution of the Langevin equations (\ref{eq:R}-\ref{eq:Y}).  The field $L_t$ defined by $L_t= -\ell R_t +Y_t$ is solution of the Langevin equation
\begin{equation*}
\partial_t L = b\Delta L -\nabla \left[\sqrt{2b}{T} (t,q) W_2 (q,t) \right]
\end{equation*}
with $b=\cfrac{1}{2} (\gamma +\gamma^{-1})$. The fields $R_t$ and $L_t$ are solutions of independent decoupled linear Langevin equations and converge as $t$ goes to infinity to independent Gaussian fields. It follows that $R_t$ and $L_t$ converge to stationary fluctuation fields ${R}_{ss}$ and ${Y}_{ss}$ such that
\begin{eqnarray*}
\text{Cov}({R}_{ss}(G),{R}_{ss}(H))=\int_{0}^{1}dq G(q)H(q){\bar T}(q)\\
\text{Cov}({Y}_{ss}(G),{Y}_{ss} (H))=\int_{0}^{1}dq G(q) H(q)\left\{{\bar T}^2 (q)+\ell^2{\bar T}(q)\right\}\\
+2(T_\ell -T_r)^{2} \int_0^1 G(q) (\Delta^{-1} H) (q) dq\\
\text{Cov}({Y}_{ss} (G),{R}_{ss}(H))=\ell\int_{0}^{1} H(q) G(q){\bar T}(q)dq
\end{eqnarray*}

Observe that the covariance of the fluctuations of energy is composed of two terms. The first one corresponds to Gaussian fluctuations for the energy under local equilibrium state while the second term represents the contribution to the covariance due to the long range correlations in the NESS. As in the case of SSEP and KMP process, the correction is given by the Green function of the Dirichlet Laplacian (\cite{BL3}, \cite{S2}).

\section{{\label{sec:6}} Large fluctuations}

\subsection{Macroscopic dynamical behavior}
Assume that initially the system is prepared in a state with a deformation profile $u_0$, energy profile ${\varepsilon}_0$ and hence temperature profile $T_0= {\varepsilon}_0 -{u}_0^2 /2$. In a diffusive scale the deformation (resp. energy, resp. temperature) profiles $u$ (resp. ${\varepsilon}$, resp. $T$) where ${\varepsilon}=T+{u^2}/2$ evolve according to the hydrodynamic equations (\ref{eq:hl}).

Our aim is to obtain the large deviation principle corresponding to the law of large numbers $(\ref{eq:hl})$. It consists to estimate the probability that the empirical quantities (deformation, energy, temperature) do not follow the corresponding solutions of $(\ref{eq:hl})$ but remain close to some prescribed paths. This probability will be exponentially small in $N$ and we look for the exponential rate. We follow the classic procedure in large deviation theory (\cite{BL2},\cite{KL}): we perturb the dynamics in such a way that the prescribed paths become typical and we compute the cost of such perturbation.\\

Fix a path ${\mathcal Y} (t, \cdot)=(u(t,\cdot),{\varepsilon} (t,\cdot))$.  The empirical deformation profile ${\mathcal R}^N_t$ and empirical energy profile ${\mathcal E}_t^N$ are defined by
\begin{eqnarray}
\label{eq:RE0}
{\mathcal R}_t^N (q)= N^{-1} \sum_{x=1}^N r_x (t) {\bf 1}_{\left[x/N, (x+1)/N \right)} (q),\\
{\mathcal E}_t^N (q)= N^{-1} \sum_{x=1}^N e_x (t) {\bf 1}_{\left[x/N, (x+1)/N \right)} (q).\nonumber
\end{eqnarray}
In appendix, we explain how to define a Markovian dynamics associated to a couple of functions $H(t,q), G(t,q)$, $q \in [0,1]$, such that the perturbed system has hydrodynamic limits given by $u$ and $\varepsilon$. This is possible if the function $F=(H,G)$ solves the Poisson equation
 \begin{equation}
\label{eq:poisson}
\begin{cases}
\partial_t {\mathcal Y}= \Delta {\mathcal Y} -\nabla(\sigma \nabla F))\\
F(t,0)=F(t,1)=(0,0)
\end{cases}
\end{equation}
where the mobility $\sigma:=\sigma (u,\varepsilon)$ is given by
\begin{equation}
\label{eq:mobility}
\sigma(u,\varepsilon)=
2\left(
\begin{array}{cc}
T & uT\\
uT & u^2 T +T^2
\end{array}
\right), \quad T={\varepsilon}-u^2 /2
\end{equation}

The perturbed process defined a probability measure $\tilde{\mathbb P}$ on the deformation/energy paths space by mean of the empirical deformation and energy profiles (see (\ref{eq:RE0})).

Our goal is to estimate the probability
\begin{eqnarray*}
\phantom{a}&\phantom{=}&{\mathbb P} \left[({\mathcal R}_s^N, {\mathcal E}_{s}^N)\sim(u(s, \cdot),\varepsilon (s, \cdot)),\; s\in [0,t]\right]\\
&=&{\tilde {\mathbb E}} \left[ \cfrac{d{\mathbb P}}{d{\tilde{{\mathbb P}}}} {\bf 1}_{\left\{({\mathcal R}_s^N, {\mathcal E}_{s}^N )\sim(r(s, \cdot),\varepsilon (s, \cdot)), \; s\in[0,t]\right\}}\right]
\end{eqnarray*}

To avoid irrelevant complications due to the fluctuations of the initial state which have no incidence on the derivation of the quasi-potential, we assume that the initial profiles $u_0$ and $T_0$ are the stationary profiles ${\bar r} (q)=\ell$ and ${\bar T} (q) =T_\ell +(T_r -T_\ell)q$. The function $F$ is such that
$${\tilde {\mathbb P}}  \left[({\mathcal R}_s^N, {\mathcal E}_{s}^N)\sim(u(s,\cdot),{\varepsilon} (s, \cdot)),\; s \in [0,t]\right]\approx 1$$
In the appendix we show that in the large $N$ limit, under ${\tilde {\mathbb P}}$, the Radon-Nikodym derivative is given by
\begin{equation*}
\cfrac{d{\mathbb P}}{d{\tilde{{\mathbb P}}}} \approx \exp\left\{ -N J_{[0,t]}(u,{\varepsilon})\right\}
\end{equation*}
where
\begin{equation}
\label{eq:29}
J_{[0,t]} (u,{\varepsilon})=\cfrac{1}{2}\int_0^{t} ds<\nabla F (s, \cdot), \sigma \nabla F (s,\cdot)>_q
\end{equation}
where $\sigma$ is here for $\sigma (u(s,\cdot),\varepsilon(s,\cdot))$ and $<\cdot,\cdot>_q$ for the usual scalar product in ${\mathbb L}^2 ([0,1],dq)$. Hence we have obtained
\begin{eqnarray*}
{\mathbb P} \left[({\mathcal R}_s^N, {\mathcal E}_{s}^N)\sim(u(s, \cdot),{\varepsilon}(s, \cdot)),\; s\in [0,t]\right] \\
\approx \exp\left\{ -N J_{[0,t]}(u,{\varepsilon})\right\}
\end{eqnarray*}

\subsection{The quasi-potential}

To understand what is the quasi-potential, consider the following situation. Assume the system is macroscopically in the stationary profile $(u(-\infty, \cdot), {\varepsilon} (-\infty,\cdot))= (\ell, {\bar T}(\cdot)+\ell^2/2)$ at $t=-\infty$ but at $t=0$ we find it in the state $(u(q),\varepsilon(q))$. We want to determine the most probable trajectory followed in the spontaneous creation of this fluctuation. According to the precedent subsection this trajectory is the one that minimizes $J_{[-\infty,0]}$ among all trajectories $({\hat u}, {\hat \varepsilon})$ connecting the stationary profiles to $(u,\varepsilon)$. The quasi-potential is then defined by
\begin{equation*}
W (u,\varepsilon) = \inf_{({\hat u}, {\hat \varepsilon})} J_{[0,t]} ({\hat u}, {\hat \varepsilon})
\end{equation*}
MFT postulates the quasi-potential $W$ is the appropriate generalization of the free energy for non-equilibrium systems and this has been proven rigorously for SSEP (\cite{BL2}). $W$ is solution of an infinite-dimensional Hamilton-Jacobi equation which is in general very difficult to solve. It has been solved for specific models (SSEP and KMP) having a single conservation law (\cite{BL2}, \cite{BL3}). For the system we consider, two quantities are conserved and  we are not able to solve this Hamilton-Jacobi equation. Nevertheless we can compute the quasi-potential for the temperature profile (\ref{eq:30}) in the case $\gamma=1$. The latter is obtained by projecting the quasi-potential $W$ on the deformation/energy profiles with a prescribed temperature profile.

Consider the system in its steady state $<\cdot>_{ss}$. Our aim is here to estimate the probability that the empirical kinetic energy defined by
\begin{equation}
\label{eq:30}
\Theta^N (q) = N^{-1} \sum_{x=1}^N p_x^2 {\bf 1}_{\left[x/N, (x+1)/N \right)} (q)
\end{equation}
is close to some prescribed temperature profile $\pi (q)$ different form the linear profile ${\bar T} (q)= T_\ell +(T_r -T_\ell)q$. This probability will be exponentially small in $N$
\begin{equation*}
\left< \left[ \Theta^N (q) \sim \pi (q) \right] \right>_{ss} \approx \exp( -N V(\pi))
\end{equation*}
By MFT, the rate function $V(\pi)$ coincides with the following \textit{projected quasi-potential}
\begin{equation*}
V(\pi)=\inf_{t >0} \inf_{(u,\varepsilon)\in \ensuremath{\mathcal A}_{t,\pi}} J_{[0,t]} (u,\varepsilon)
\end{equation*}
where the paths set ${\ensuremath{\mathcal A}}_{t,\pi}$ is defined by
\begin{eqnarray*}
{\ensuremath{\mathcal A}}_{t,\pi} = \left\{ (u,\varepsilon); \quad {\varepsilon}(t, \cdot)-\cfrac{u^2(t, \cdot)}{2}=\pi(\cdot); \right.\\
\left. {\phantom{\cfrac{u^2(t, \cdot)}{2}}}u (0,\cdot)=\ell,\, T(0,\cdot)= {\bar T}(\cdot)\right\}
\end{eqnarray*}
Paths ${\mathcal Y}=(u,\varepsilon) \in {\mathcal A}_{t,\pi}$ must also satisfy the boundary conditions
\begin{equation}
\label{eq:bc}
u(t,0)=u(t,1)=\ell, \quad \varepsilon (t,0)=T_\ell+\ell^2/2, \, \varepsilon (t,1)=T_r +\ell^2/2
\end{equation}
In fact, it can be shown that $J_{[0,t]} (u,\varepsilon) =+\infty$ if the path ${\mathcal Y}$ does not satisfy these boundary conditions.\\

Our main result is the computation of the projected quasi-potential:
\begin{equation}
\label{eq:32}
V (\pi)= \inf_{\tau \in \ensuremath{\mathcal T}} [{\ensuremath{\mathcal F}} (\pi, \tau) ]
\end{equation}
where $\ensuremath{\mathcal T}=\{ \tau \in C^{1}([0,1]); \; \tau' (q)>0,\; \tau(0)=T_\ell,\; \tau (1)=T_r\}$ and
\begin{equation*}
{\cal F} (\pi,\tau)= \int_0^1 dq \left[\cfrac{\pi (q)}{\tau(q)} -1 - \log\cfrac{\pi(q)}{\tau (q)} -\log \cfrac{\tau' (q)}{(T_r - T_{\ell})}\right]
\end{equation*}

Before proving (\ref{eq:32}) let us make some remarks. First, $V(\pi)$ is equal to the rate function for the KMP process (\cite{BL3}). Nevertheless, it is not easy to understand the deep reason. The symmetric part ${\mathcal S}$ of the generator ${\mathcal L}$ is more or less a time-continuous version of the KMP process for the kinetic energy but the Hamiltonian part has a non-trivial effect on the latter since it mixes momenta with positions. Hence, the derivation of the quasi-potential for the kinetic energy can not be derived from the the computations for the KMP process. Secondly, we are able to compute $V$ only for $\gamma=1$. When $\gamma$ is equal to $1$ hydrodynamic equations for the deformation and for the energy are decoupled but since temperature is a non-linear function of deformation and energy, it is not clear why it helps-- but it does.
Formula (\ref{eq:32}) shows that the large deviation functional $V$ is nonlocal and consequently not additive: the probability of temperature profile in disjoint macroscopic regions is not given by the product of the separate probabilities. Nonlocality is a generic feature of NESS and is related to the ${\mathcal O} (N^{-1})$ corrections to local thermal equilibrium.

\vspace{0,5cm}
Let us call $S(\pi)$ the right hand side of equality (\ref{eq:32}).
\vspace{0,5cm}

For every time independent deformation/energy profiles $(r(q),e(q))$  and $\tau (q) \in \ensuremath{\mathcal T}$ we define the functional
\begin{equation}
\label{eq:U}
U(r,e,\tau)=\int_0^1 dq \left\{ \cfrac{T}{\tau} -1 - \log \cfrac{T}{\tau} -\log \cfrac{\tau'}{T_\ell - T_r} +\cfrac{(r-\ell)^2}{2\tau}\right\}
\end{equation}
where $T(q)=e(q)-r(q)^2/2$ the temperature profile corresponding to $(r(q),e(q))$. Define the function $\tau:=\tau(r,e)$ of ${\ensuremath{\mathcal T}}$ as the unique increasing solution of:
\begin{equation}
\label{eq:tau}
\begin{cases}
\tau^2 \cfrac{\Delta \tau}{(\nabla \tau)^2} = \tau -T -\cfrac{1}{2} (r-\ell)^2\\
\tau(0)=T_\ell,\; \tau (1)= T_r
\end{cases}
\end{equation}
Fix deformation/energy paths satisfying boundary conditions (\ref{eq:bc}) and define ${\mathcal Z}$ by
\begin{equation}
\label{eq:YZ}
{\mathcal Y}=\left(\begin{array}{c}u\\\varepsilon \end{array}\right),\qquad {\mathcal Z}=\left[\partial_t {\mathcal Y}-\Delta {\mathcal Y}+\nabla(\sigma \nabla(\delta U))\right],\\
\end{equation}
In the appendix we show the following formula
\begin{widetext}
\begin{eqnarray}
\label{eq:43}
J_{[0,t]}(u,\varepsilon)=U(u(t,\cdot),{\varepsilon} (t,\cdot),\tau({\varepsilon} (t,\cdot),u(t,\cdot)))-U(u(0, \cdot),{\varepsilon}(0,\cdot),\tau(u(0,\cdot),{\varepsilon} (0,\cdot)))\\
\nonumber \\
+\cfrac{1}{2}\int_0^t ds \left<\nabla^{-1}{\mathcal Z}, \sigma^{-1}\nabla^{-1}{\mathcal Z} \right>_q+\cfrac{1}{4}\int_0^t ds  \int_0^1 dq (u(s,q)-\ell)^4 \cfrac{(\nabla \tau)^2 (s,q)}{\tau^4 (s,q)}\nonumber
\end{eqnarray}
\end{widetext}
where
\begin{equation*}
\delta U=\left(\begin{array}{c}\cfrac{\delta U}{\delta r}\\ \phantom{a}\\ \cfrac{\delta U}{\delta {e}} \end{array}\right) (u,\varepsilon,\tau(u,\varepsilon))
\end{equation*}

If $(u,\varepsilon)$ belongs to ${\mathcal A}_{t,\pi}$,
\begin{eqnarray*}
U(\, u(0, \cdot),{\varepsilon}(0,\cdot),\tau(u(0,\cdot),{\varepsilon} (0,\cdot))\,)\\
=U(\,\ell, \bar{T}+\ell^2/2,\tau(\ell, {\bar T} +\ell^2 /2)\,)=0
\end{eqnarray*}
and
$$U(u(t, \cdot),{\varepsilon} (t,\cdot), \tau(u(t, \cdot), \varepsilon(t, \cdot))) \ge {\mathcal F} (\pi, \tau (u(t,\cdot),{\varepsilon} (t,\cdot)).$$
The two last terms on the right hand side of (\ref{eq:43}) are positive so that for every paths in ${\mathcal A}_{t,\pi}$, we have
\begin{eqnarray*}
J_{[0,t]}(u,\varepsilon) \geq S(\pi)
\end{eqnarray*}
and we obtain hence
\begin{equation}
\label{eq:firstinequality}
V(\pi)\geq S(\pi)
\end{equation}

To obtain the other sense of the inequality, we have to construct an optimal path $(u^*,{\varepsilon}^*) \in \ensuremath{\mathcal A}_{t,\pi}$ such that the two last terms in the right hand side of $(\ref{eq:43})$ are equal to $0$, i.e.
\begin{equation}
\label{eq:44}
\begin{cases}
\partial_t {\mathcal Y}=\Delta {\mathcal Y}-\nabla(\sigma \nabla(\delta U))\\
u(t,q)=\ell
\end{cases}
\end{equation}
We note $T^{*}={\varepsilon}^* -{u^*}^2/2$ the corresponding temperature. Then reporting in (\ref{eq:43}), we obtain
\begin{equation}
\label{eq:10000}
J_{[0,t]}(u^*,{\varepsilon}^*)=U(u^{*}(t),{\varepsilon}^{*}(t),\tau(u^*(t) , {\varepsilon}^*(t)))
\end{equation}
By the definition (\ref{eq:U}) of $U$ and by using the fact that $u^{*} (t,q)=\ell$, we obtain
\begin{eqnarray}
\label{eq:10001}
J_{[0,t]}(u^*,{\varepsilon}^*)={\ensuremath{\mathcal F}}(T^{*}(t, \cdot),\tau(u^*(t, \cdot), {\varepsilon}^{*}(t, \cdot)) \nonumber\\
={\ensuremath{\mathcal F}}(\pi,\tau(\ell,\pi +\ell^2 /2))
\end{eqnarray}
The variational problem defining $S$ is solved for $\tau= \tau(\ell,\pi+\ell^2 /2)$ (\cite{BL3}) so that
$$S(\pi)={\ensuremath{\mathcal F}} (\pi, \tau(\ell,\pi +\ell^2 /2))$$
and therefore we have
\begin{equation*}
V(\pi)=\inf_{t >0} \inf_{\ensuremath{\mathcal A}_{t,\pi}} J_{[0,t]}(u,\varepsilon) \leq S(\pi)
\end{equation*}
This inequality with (\ref{eq:firstinequality}) shows that $V(\pi)=S(\pi)$.
It remains to prove that such ``good'' path exists. The proof is similar to \cite{BL3} and we shall merely outline it. Equation (\ref{eq:44}) is equivalent to the following one
\begin{equation}
\label{eq:45}
\begin{cases}
\begin{array}{l}
\partial_t T^{*}=-\Delta(T^*)+2\nabla\left[\cfrac{(T^*)^2}{(\tau^*)^2}\nabla(\tau^*)\right]\\
u^*(t,q)=\ell
\end{array}
\end{cases}
\end{equation}
where $\tau^*(t,\cdot)= \tau(\ell, T^*(t,\cdot) +\ell^2/2)$. Let us denote by $\theta^*(s, \cdot)=T^{*}(t-s, \cdot)$ the time reversed path of $T^*$. $\theta^{*}$ can be constructed in the following procedure. We define $\theta^{*}(s,q),\; s\in[0,t],\; q\in [0,1]$ by
$$\theta^{*}(s, \cdot)= \rho (s, \cdot) -2\rho(s, \cdot)^2 \cfrac{\Delta \rho (s, \cdot)}{[(\nabla \rho)(s, \cdot)]^2}$$
where $\rho(s,q)$ is the solution of
\begin{equation*}
\label{eq47}
\begin{cases}
\partial_s \rho =\Delta \rho\\
\rho (s,0)=T_{\ell},\qquad \rho (s,1)=T_r\\
\rho(0,q)=\rho_0(q)=\tau(\ell, \pi+\ell^2/2)(q)
\end{cases}
\end{equation*}
It can be checked that $T^{*}(s,q)=\theta^{*}(t-s,q)$ solves $(\ref{eq:45})$. Moreover, we have $T^{*}(0,\cdot)=\theta^{*}(t,\cdot)$ and $T^{*}(t,\cdot)=\pi(\cdot)$. This path belongs to ${\ensuremath{\mathcal A}}_{t,\pi}$ only as $t\to \infty$ since $\theta^{*}(t,\cdot)$ goes to $\bar{T}(\cdot)$ as $t\to +\infty$. We have hence in fact to define $T^*$ by the preceding procedure in some time interval $[t_1,t]$ and to interpolate $\bar{T}(\cdot)$ to $\pi^{*}(t_1, \cdot)$ in the time interval $[0,t_1]$ (see \cite{BL2}, \cite{BL3} for details).
This optimal path is also obtained as the time reversed solution of the hydrodynamic equation corresponding to the process with generator ${\ensuremath{\mathcal L}}^{*}$. It is easy to show that this last hydrodynamic equation is in fact the same as the hydrodynamic equation corresponding to ${\ensuremath{\mathcal L}}$. This is the "generalized" Onsager-Machlup theory developed in \cite{BL1} for NESS: "the spontaneous emergence of a macroscopic fluctuation takes place most likely following a trajectory which can be characterized in terms of the time reversed process."  Observe also the following a priori non trivial fact: the optimal path is obtained with a constant deformation profile.

\section{Conclusions}

In the present work we obtained hydrodynamic limits, Gaussian fluctuations and (partially) large fluctuations for a model of harmonic oscillators perturbed by a conservative noise. Up to now MFT has been restricted to gradient systems with a single conservation law. This work is hence the first one where MFT is applied for a non-gradient model with two conserved quantities. The quasi-potential for the temperature has been computed in the case $\gamma=1$ and it turns out that it coincides with the one of the KMP process. Our results show this system exhibits generic features of non-equilibrium models : long range correlations and non-locality of the quasi-potential.

Nevertheless our study is not completely satisfactory. It would be interesting to extend the previous results to the case $\gamma \neq 1$ and to compute the quasi-potential for the two conserved quantities and not only for the temperature. The difficulty is that there does not exist general strategy to solve the corresponding infinite-dimensional Hamilton-Jacobi equation.


\section{Introduction}

The uncertainty principle in harmonic analysis is a class
of theorems which state that a nontrivial function
and its Fourier transform can not both be too sharply localized.
For background on different appropriate notions of localization
and an overview on the recent renewed interest in mathematical formulations
of the uncertainty principle, see the survey \cite{JP.FS}.
This paper will adopt the broader view
that the uncertainty principle can be seen not only as a statement
about the time-frequency localization of a single function but also as a
statement on the degradation of localization when one considers successive elements
of an orthonormal basis.  In particular, the results that we consider
show that the elements of an orthonormal basis as well as their Fourier transforms
can not be uniformly concentrated in the time-frequency plane.

Hardy's Uncertainty Principle \cite{JP.Ha} may be viewed as an early theorem of this type.
To set notation, define the
\emph{Fourier transform} of $f\in L^1({\mathbb{R}})$ by
$$
\widehat{f}(\xi)=\int f(t)e^{-2i\pi t\xi} {d}t,
$$
and then extend to $L^2({\mathbb{R}})$ in the usual way.

\medskip

\begin{theorem} [Hardy's Uncertainty Principle]   Let $a,b,C,N>0$ be positive real numbers and let
$f \in L^2({\mathbb{R}})$. Assume that for almost every $x,\xi \in {\mathbb{R}}$,
\begin{equation} \label{eq_hardy}
|f(x)|\leq C(1+|x|)^Ne^{-\pi a|x|^2}\ \ \  \hbox{ and } \ \ \
|\widehat{f}(\xi)|\leq C(1+|\xi|)^Ne^{-\pi b|\xi|^2}.
\end{equation}
The following hold:
\begin{itemize}
\item If $ab>1$ then $f=0$.
\item If $ab=1$ then $f(x)=P(x)e^{-\pi a|x|^2}$ for some polynomial $P$ of degree at most $N$.
\end{itemize}
\end{theorem}

This theorem has been further generalized where the pointwise condition (\ref{eq_hardy})
is replaced by
integral conditions in \cite{JP.BDJ}, and by distributional conditions in \cite{JP.De}.
Also see \cite{AP.GZ} and \cite{AP.HL}.
One may interpret Hardy's theorem by saying that the set of functions which, along their
Fourier transforms, is bounded by $C(1+|x|)^Ne^{-\pi|x|^2}$ is finite dimensional,
in the sense that its
span is a finite dimensional subspace of $L^2({\mathbb{R}})$.

In the case $ab<1$, the class of functions satisfying the condition (\ref{eq_hardy})
has been fully described by B. Demange \cite{JP.De}.
In particular, it is an infinite dimensional subset of $L^2({\mathbb{R}})$.
Nevertheless, it can not contain an infinite orthonormal sequence.
Indeed, this was first proved by Shapiro in \cite{JP.Sh}:

\begin{theorem} [Shapiro's Umbrella Theorem] \label{ap.sh-umbrel}
Let $\varphi,\psi \in  L^2({\mathbb{R}})$.  If
$\{ e_k \} \subset L^2 ({\mathbb{R}})$ is an orthonormal sequence of functions such that for all $k$ and for almost all $x,\xi\in{\mathbb{R}}$,
$$
|e_k(x)|\leq |\varphi(x)|\quad\mbox{and}\quad|\widehat{e}_k(\xi)|\leq |\psi(\xi)|,
$$
then the sequence $\{e_k\}$ is finite.
\end{theorem}

Recent work of A. De Roton, B. Saffari, H.S. Shapiro, G. Tennenbaum,
see \cite{JP.DeR}, shows that the assumption $\varphi,\psi \in L^2 ({\mathbb{R}})$ can not be substantially weakened.
Shapiro's elegant proof of Theorem \ref{ap.sh-umbrel} uses a compactness argument of Kolmogorov,
see \cite{JP.Sh2},
but does not give a bound on the number of elements in the finite sequence.

A second problem of a similar nature studied by Shapiro in \cite{JP.Sh} is that
of bounding the means and variances of orthonormal sequences.
For $f\in L^2({\mathbb{R}})$ with $\norm{f}_2=1$, we define the following associated
\emph{mean}
$$
\mu(f)={\rm Mean}(|f|^2) = \int t|f(t)|^2\mbox{d}t,
$$
and the associated \emph{variance}
$$
\Delta^2(f)= {\rm Var} (|f|^2) = \int|t-\mu(f)|^2|f(t)|^2\mbox{d}t.
$$
It will be convenient to work also with the \emph{dispersion} $\Delta(f)\equiv\sqrt{\Delta^2(f)}$.
In \cite{JP.Sh}, Shapiro posed the question of determining for which sequences of real numbers
$\{a_n\}_{n=0}^{\infty},$ $\{b_n\}_{n=0}^{\infty},$ $\{c_n\}_{n=0}^{\infty},$
$\{d_n\}_{n=0}^{\infty} \subset {\mathbb{R}}$ there exists an orthonormal basis
$\{e_n\}_{n=0}^{\infty}$ for $L^2 ({\mathbb{R}})$ such that for all $n \geq 0$
$$
\mu(e_n)=a_n,\ \mu(\widehat{e}_n)=b_n,\ \Delta(e_n)=c_n,\ \Delta(\widehat{e}_n)=d_n.
$$
Using again Kolmogorov's compactness argument,
he proved the following, \cite{JP.Sh}:

\begin{theorem} [Shapiro's Mean-Dispersion Principle]\label{ap.sh-meandisp}
There does not exist an infinite orthonormal sequence $\{e_n\}_{n=0}^{\infty}
\subset L^2({\mathbb{R}})$ such that
all four of $\mu(e_n)$, $\mu(\widehat{e}_n)$, $\Delta(e_n), \Delta(\widehat{e}_n)$ are uniformly bounded.
\end{theorem}

\medskip

An extension of this theorem in \cite{JP.Po} shows that if
$\{e_n\}_{n=0}^{\infty}$ is an orthonormal basis for $L^2({\mathbb{R}})$ then
two dispersions and one mean $\Delta(e_n), \Delta(\widehat{e}_n),\mu(e_n)$
can not all be uniformly bounded.
Shapiro recently pointed out a nice alternate proof of this result using
the Kolmogorov compactness theorem from \cite{JP.Sh}.  The case for two means and one dispersion
is different.   In fact, it is possible to construct an
orthonormal basis $\{e_n\}_{n=0}^{\infty}$ for $L^2({\mathbb{R}})$ such that
the two means and one dispersion $\mu(e_n)$, $\mu(\widehat{e}_n),\Delta(e_n)$
are uniformly bounded, see \cite{JP.Po}.

Although our focus will be on Shapiro's theorems,
let us also briefly refer the reader to some other work in the literature concerning
uncertainty principles for bases.
The classical Balian-Low theorem states that if a set of lattice coherent states forms
an orthonormal basis for $L^2({\mathbb{R}})$ then the window function satisfies a strong
version of the uncertainty principle, e.g., see \cite{AP.CP,AP.GHHK}.
For an analogue concerning dyadic orthonormal wavelets, see \cite{AP.battle}.

\subsection*{Overview and main results}
The goal of this paper is to provide quantitative versions of Shapiro's
Mean-Dispersion Principle and Umbrella
Theorem, i.e., Theorems \ref{ap.sh-umbrel} and \ref{ap.sh-meandisp}.

Section \ref{sec2.ap} addresses the Mean-Dispersion Theorem.
The main results of this section are contained in
Section \ref{ap.sharp.md.sec} where we prove a sharp quantitative
version of Shapiro's Mean-Dispersion Principle.
This result is sharp, but the method of proof
is not easily applicable to more general
versions of the problem.
Sections \ref{ap.herm.sec} and \ref{ap.rayleigh.sec} respectively
contain necessary background on Hermite functions and the Rayleigh-Ritz technique which is needed
in the proofs.  Section \ref{ap.riesz.shmd.sec} proves a version of the mean-dispersion theorem for Riesz bases.

Section \ref{sec3.ap} addresses the Umbrella Theorem
and variants of the Mean-Dispersion Theorem.
The main results of this section are contained in
Section \ref{gen.md.ap} where we prove a
quantitative version of the Mean-Dispersion Principle for a generalized
notion of dispersion, and in Section \ref{JP.sec:QUT} where
we prove a quantitative version of Shapiro's Umbrella Theorem.
Explicit bounds on the size of possible orthonormal sequences are given in particular cases.
Since the methods of Section \ref{sec2.ap} are no longer
easily applicable here, we adopt an approach based on geometric combinatorics.
Our results use estimates on the
size of spherical codes, and the theory of prolate sphero\"\i dal wavefunctions.
Section \ref{ap.sphcode.sec} contains background results on spherical codes,
including the Delsarte, Goethals, Seidel bound.
Section \ref{ap.apponb.sec} proves some necessary results on projections of one set of orthonormal functions onto
another set of orthonormal functions.  Section \ref{ap.psw.sec} gives an overview of the prolate sphero\"\i dal wavefunctions
and makes a connection between projections of orthonormal functions and spherical codes.
Section \ref{ap.rieszang.sec} concludes with extensions to Riesz bases.

\section{Growth of means and dispersions} \label{sec2.ap}

In this section, we use the classical Rayleigh-Ritz technique to give a quantitative version of
Shapiro's Mean-Dispersion Theorem. We also prove that, in this sense, the Hermite
basis is the best concentrated orthonormal basis of $L^2({\mathbb{R}})$.

\subsection{The Hermite basis} \label{ap.herm.sec}

Results of this section can be found in \cite{JP.FS}.
The Hermite functions are defined by
$$
h_k(t)=\frac{2^{1/4}}{\sqrt{k!}}\left(-\frac{1}{\sqrt{2\pi}}\right)^ke^{\pi t^2} \left(\frac{{d}}{{d}t}\right)^ke^{-2\pi t^2}.
$$
It is well known that the Hermite functions are eigenfunctions of the Fourier transform,
satisfy $\widehat{h_k}=i^{-k}h_k$, and
form an orthonormal basis for $L^2({\mathbb{R}})$.
Let us define the {\em Hermite operator} $H$ for
functions $f$ in the Schwartz class by
$$Hf(t) = -\frac{1}{4\pi^2}\frac{d^2}{dt^2}f(t) + t^2 f(t).$$
It is easy to show that
\begin{equation}
\label{JP.eigen_val}
Hh_k = \left( \frac{2k+1}{2\pi} \right) h_k,
\end{equation}
so that $H$ may also be seen as the densely defined,
positive, self-adjoint, unbounded operator on $L^2(\mathbb{R})$ defined by
$$
Hf = \sum_{k=0}^{\infty} \frac{2k+1}{2\pi} \langle f, h_k \rangle h_k.
$$
From this, it immediately follows that, for each $f$ in the domain of $H$
\begin{eqnarray}
\langle Hf,f \rangle&=&
\sum_{k=0}^{\infty} \frac{2k+1}{2\pi} |\langle f, h_k \rangle |^2
= \int |t|^2 |f(t)|^2 dt + \int |\xi|^2 |\widehat{f}(\xi)|^2 d\xi \label{JP.eq:fundherm} \\
&=&\mu(f)^2\norm{f}_2^2+\Delta^2(f)+\mu(\widehat{f})^2\|\widehat{f}\|_2^2+\Delta^2(\widehat{f}).
\notag
\end{eqnarray}

\subsection{The Rayleigh-Ritz Technique} \label{ap.rayleigh.sec}

The Rayleigh-Ritz technique
is a useful tool for estimating eigenvalues of operators, see \cite[Theorem XIII.3, page 82]{JP.RS4}.

\begin{theorem} [The Rayleigh-Ritz Technique]
\label{JP.ral_ritz_thm}
Let $H$ be a positive self-adjoint operator and define
$$
\lambda_k (H) = \sup_{\varphi_0, \cdots, \varphi_{k-1}} \ \
\inf_{\psi \in [\varphi_0, \cdots, \varphi_{k-1}]^{\perp},\|\psi\|_2 \leq 1,
\psi \in D(H)}
\langle H \psi, \psi \rangle.$$
where $D(H)$ is the domain of $H$. Let $V$ be a $n+1$ dimensional
subspace, $V\subset D(H),$ and let $P$ be the orthogonal projection onto $V$.
Let $H_V = PHP$ and let $\widetilde{H_V}$ denote the restriction of
$H_V$ to $V$.
Let $\mu_0 \leq \mu_1 \leq \cdots \leq \mu_n$ be the eigenvalues of
$\widetilde{H_V}$
Then
$$
\lambda_k (H) \leq \mu_k, \ \ \ k=0, \cdots, n.
$$
\end{theorem}

The following corollary is a standard and useful application
of the Rayleigh-Ritz technique.  For example, \cite[Chapter 12]{JP.LL}
contains a version in the setting of Schr\"odinger operators.

\begin{corollary}
\label{JP.trace_cor}
Let $H$ be a positive self-adjoint operator, and
let $\varphi_0, \cdots, \varphi_n$ be an orthonormal set of functions.  Then
\begin{equation}
\label{JP.eq:trace}
\sum_{k=0}^n \lambda_k(H)  \leq \sum_{k=0}^n
\langle H\varphi_k, \varphi_k \rangle.
\end{equation}
\end{corollary}

\begin{proof}
If some $\varphi_k \notin D(H)$ then positivity of $H$ implies that \eqref{JP.eq:trace} trivially
holds since the right hand side of the equation would be infinite.
We may thus assume that $\varphi_0, \cdots, \varphi_n\in D(H)$.

Define the $n+1$ dimensional subspace $V = {\rm span} \ \{\varphi_k\}_{k=0}^n$
and note that the operator $\widetilde{H_V}$
is given by the matrix
$M =[\langle H \varphi_j, \varphi_k \rangle]_{0\leq j,k \leq n}$.
Let $\mu_0, \cdots, \mu_n$ be the eigenvalues of $\widetilde{H_V}$,
i.e., of the matrix $M$.
By Theorem \ref{JP.ral_ritz_thm},
$$
\sum_{k=0}^n \lambda_k (H) \leq \sum_{k=0}^n \mu_k = {\rm Trace} (M)
= \sum_{k=0}^n \langle H\varphi_k, \varphi_k \rangle
$$
which completes the proof of the corollary.
\end{proof}

\subsection{The Sharp Mean-Dispersion Principle} \label{ap.sharp.md.sec}

\begin{theorem} [Mean-Dispersion Principle]
\label{JP.th:optherm}
Let $\{e_k\}_{k=0}^{\infty}$ be any orthonormal sequence in $L^2({\mathbb{R}})$.
Then for all $n\geq 0$,
\begin{equation}
\label{JP.eq:optherm}
\sum_{k=0}^n \left( \Delta^2(e_k) + \Delta^2(\widehat{e_k}) +
|\mu(e_k)|^2 + |\mu(\widehat{e_k})|^2 \right) \geq
\frac{(n+1)(2n+1)}{4\pi}.
\end{equation}
Moreover, if equality holds for all $0 \leq n\leq n_0$, then there exists $\{c_k\}_{n=0}^{n_0}
\subset {\mathbb{C}}$ such that $|c_k|=1$ and $e_k=c_kh_k$ for each $0 \leq k \leq n_0$.
\end{theorem}

\begin{proof}
Since $H$ is positive and self-adjoint, one may use Corollary
\ref{JP.trace_cor}.  It follows from Corollary \ref{JP.trace_cor} that for each
$n \geq 0$ one has
\begin{equation}
\label{JP.trace_eq}
\sum_{k=0}^n \frac{2k+1}{2\pi} \leq
\sum_{k=0}^n \langle He_k, e_k \rangle.
\end{equation}
From (\ref{JP.eq:fundherm}), note that since $\|e_k\|_2=1$,
$$
\langle H e_k, e_k \rangle  =
\Delta^2 (e_k) + \Delta^2 (\widehat{e_k}) + |\mu(e_k)|^2
+ |\mu(\widehat{e_k})|^2.
$$
This completes the proof of the first part.

Assume equality holds in (\ref{JP.eq:optherm}) for all $n=0,\ldots,n_0$,
in other terms that, for $n=0,\ldots,n_0$,
$$
\scal{He_n,e_n}=
\Delta^2(e_n) + \Delta^2(\widehat{e_n}) +
|\mu(e_n)|^2 + |\mu(\widehat{e_n})|^2 =
\frac{2n+1}{2\pi}.
$$
Let us first apply (\ref{JP.eq:fundherm}) for $f=e_0$:
$$
\sum_{k=0}^{\infty} \frac{2k+1}{2\pi} |\langle e_0, h_k \rangle |^2
=\scal{He_0,e_0}=\frac{1}{2\pi}
=\sum_{k=0}^{\infty} \frac{1}{2\pi} |\langle e_0, h_k \rangle |^2
$$
since $\|e_0\|_2=1$.
Thus, for $k\geq1$, one has $\scal{e_0,h_k}=0$ and hence $e_0=c_0h_0$.
Also $\|e_0\|_2=1$ implies $|c_0|=1$.
Next, assume that we have proved $e_k=c_kh_k$
for $k=0,\ldots,n-1$.
Since $e_n$ is orthogonal to $e_k$ for $k<n,$ one has $\scal{e_n,h_k}=0$.
Applying (\ref{JP.eq:fundherm}) for $f=e_n$
we obtain that,
$$
\sum_{k=n}^{\infty} \frac{2k+1}{2\pi} |\langle e_n, h_k \rangle |^2
=\sum_{k=0}^{\infty} \frac{2k+1}{2\pi} |\langle e_n, h_k \rangle |^2
=\scal{He_n,e_n}=\frac{2n+1}{2\pi}
=\sum_{k=n}^{\infty} \frac{2n+1}{2\pi} |\langle e_n, h_k \rangle |^2.
$$
Thus $\scal{e_n,h_k}=0$ for $k>n$. It follows that $e_n=c_nh_n$.
\end{proof}

\begin{examplenum}
For all $n \geq 0$, the Hermite functions satisfy
$$\mu(h_n) = \mu(\widehat{h_n}) =0 \ \ \ \hbox{ and } \ \ \
\Delta^2 (h_n) =\Delta^2 (\widehat{h_n}) = \frac{2n+1}{4\pi}.$$
\end{examplenum}
For comparison, let us remark that
Bourgain has constructed an orthonormal basis $\{b_n\}_{n=1}^{\infty}$
for $L^2({\mathbb{R}})$, see \cite{JP.B},
which
satisfies $\Delta^2 (b_n) \leq \frac{1}{2\sqrt{\pi}} + \varepsilon$
and $\Delta^2 (\widehat{b_n}) \leq \frac{1}{2\sqrt{\pi}} + \varepsilon$.  However,
it is difficult to control the growth of $\mu(b_n), \mu(\widehat{b_n})$ in this
construction.
For other bases with more structure, see the
related work in \cite{AP.BCGP} that constructs
an orthonormal basis of lattice coherent states $\{g_{m,n}\}_{m,n\in{\mathbb{Z}}}$
for $L^2({\mathbb{R}})$
which is logarithmically close to having
uniformly bounded dispersions.  The means
$(\mu(g_{m,n}), \mu( \widehat{g_{m,n}}))$ for this basis lie on a translate of
the lattice ${\mathbb{Z}} \times {\mathbb{Z}}$.

It is interesting to note that if one takes $n=0$ in Theorem \ref{JP.th:optherm} then this yields the usual
form of Heisenberg's uncertainty principle (see \cite{JP.FS} for
equivalences between uncertainty principles with sums and products).
In fact, using (\ref{JP.eq:fundherm}), Theorem \ref{JP.th:optherm}
also implies a more general version of Heisenberg's uncertainty
principle that is implicit in \cite{JP.FS}.
In particular, if $f\in L^2({\mathbb{R}})$ with $\norm{f}_2=1$ is orthogonal to $h_0,\ldots,h_{n-1}$ then
$$
\Delta^2(f) + \Delta^2(\widehat{f}) +
|\mu(f)|^2 + |\mu(\widehat{f})|^2 \geq \frac{2n+1}{2\pi}.
$$

For instance, if $f$ is odd, then $f$ is orthogonal to $h_0$,
and $\mu(f)=\mu(\widehat{f})=0$. Using the usual scaling trick,
we thus get the well known fact that
the optimal constant in Heisenberg's inequality, e.g., see \cite{JP.FS}, is given as follows
$$
\Delta(f)\Delta(\widehat{f}) \geq
\begin{cases}
\frac{1}{4\pi}\norm{f}_2^2&\mbox{in general}\\
\frac{3}{4\pi}\norm{f}_2^2&\mbox{if $f$ is odd}
\end{cases}.
$$

\begin{corollary} \label{ap.cor.opt.md}
Fix $A>0.$  If $\{e_k\}_{k=0}^{n} \subset L^2({\mathbb{R}})$ is an orthonormal sequence and
for $k=0,\ldots,n$, satisfies
$$
|\mu(e_k)|,\ |\mu(\widehat{e_k})|,\
\Delta(e_k),\ \Delta(\widehat{e_k})\leq A,
$$
then
$n\leq 8\pi A^2$.
\end{corollary}

\begin{proof}
According to Theorem \ref{JP.th:optherm}
$$
4(n+1)A^2\geq
\sum_{k=0}^n \left( \Delta^2(e_k) + \Delta^2(\widehat{e_k}) +
|\mu(e_k)|^2 + |\mu(\widehat{e_k})|^2 \right) \geq \frac{(n+1)(2n+1)}{4\pi}.
$$
It follows that $2n+1\leq 16\pi A^2$.
\end{proof}

This may also be stated as follows:

\begin{corollary}
\label{JP.cor:cor2}
If $\{e_k\}_{k=0}^{\infty} \subset L^2({\mathbb{R}})$ is an orthonormal sequence, then
for every $n$,
$$
\max\{|\mu(e_k)|,\ |\mu(\widehat{e_k})|,\ \Delta(e_k),\ \Delta(\widehat{e_k})\ : 0 \leq k \leq n\}
\geq \sqrt{\frac{2n+1}{16\pi}}.
$$
\end{corollary}

\subsection{An extension to Riesz bases} \label{ap.riesz.shmd.sec}

Recall that $\{x_k\}_{k=0}^{\infty}$ is a
\emph{Riesz basis} for $L^2({\mathbb{R}})$ if there
exists an isomorphism, $U:L^2({\mathbb{R}})\to L^2({\mathbb{R}})$, called the \emph{orthogonalizer}
of $\{x_k\}_{k=0}^{\infty}$, such that
$\{Ux_k\}_{k=0}^{\infty}$ is an orthonormal basis for $L^2({\mathbb{R}})$.
It then follows that, for every
$\{a_n\}_{n=0}^{\infty} \in\ell^2$,
\begin{equation}
\frac{1}{\norm{U}^2}\sum_{n=0}^{\infty} |a_n|^2
\leq \norm{\sum_{n=0}^{\infty} a_nx_n}_{2}^2\leq\norm{U^{-1}}^2\sum_{n= 0}^{\infty} |a_n|^2.
\label{JP.eq:riesz}
\end{equation}

One can adapt the results of the previous sections to Riesz bases.
To start, note that the Rayleigh-Ritz technique leads to the
following, cf. \cite[Theorem XIII.3, page 82]{JP.RS4}:

\begin{lemma}
\label{JP.lem:traceR}
Let $H$ be a positive, self-adjoint, densely defined operator on $L^2 ({\mathbb{R}})$,
and let $\{x_k\}_{k=0}^{\infty}$ be a Riesz basis for $L^2({\mathbb{R}})$ with
orthonormalizer $U$. Then, for every $n\geq 0$,
\begin{equation}
\label{JP.eq:traceR}
\sum_{k=0}^n \lambda_k(H)  \leq \norm{U}^2 \sum_{k=0}^n
\langle Hx_k, x_k \rangle.
\end{equation}
\end{lemma}

\begin{proof} Let us take the notations of the proof of Corollary \ref{JP.trace_cor}.
Write $\varphi_k=Ux_k$, it is then enough to notice that
$$
M=[\scal{Hx_k,x_k}]=[\scal{HU^{-1}x_k,U^{-1}x_k}]=[\scal{U^{-1}\,^*HU^{-1}x_k,x_k}].
$$
As $U^{-1}\,^*HU^{-1}$ is a positive operator, that the Rayleigh-Ritz theorem gives
$$
\sum_{k=0}^n\langle Hx_k, x_k \rangle\geq \sum_{k=0}^n\lambda_k(U^{-1}\,^*HU^{-1}).
$$
But,
\begin{eqnarray*}
\lambda_k(U^{-1}\,^*HU^{-1})&=&\sup_{\varphi_0, \cdots, \varphi_{k-1}} \ \
\inf_{\psi \in [\varphi_0, \cdots, \varphi_{k-1}]^{\perp},\|\psi\|_2\leq1}
\scal{U^{-1}\,^*HU^{-1}\psi,\psi}\\
&=& \sup_{\varphi_0, \cdots, \varphi_{k-1}}
\inf_{\psi \in [\varphi_0, \cdots, \varphi_{k-1}]^{\perp},\|\psi\|_2\leq1}
\scal{HU^{-1}\psi,U^{-1}\psi}\\
&=& \sup_{\varphi_0, \cdots, \varphi_{k-1}}
\inf_{\tilde\psi \in [U^*\varphi_0, \cdots, U^*\varphi_{k-1}]^{\perp},\|U\tilde\psi\|_2\leq1}
\scal{H\tilde\psi,\tilde\psi}
\end{eqnarray*}
and, as $\|U\tilde\psi\|_2\leq \norm{U}\,\|\tilde\psi\|_2$,
$$
\lambda_k(U^{-1}\,^*HU^{-1})\geq \frac{1}{\norm{U}^2}
\sup_{\tilde\varphi_0, \cdots, \tilde\varphi_{k-1}}
\inf_{\tilde\psi \in [\tilde\varphi_0, \cdots, \tilde\varphi_{k-1}]^{\perp},\|\tilde\psi\|_2\leq1}
\scal{H\tilde\psi,\tilde\psi}=\frac{1}{\norm{U}^2}\lambda_k(H).
$$
\end{proof}

Adapting the proofs of the previous section, we obtain the following corollary.

\begin{corollary}
If $\{ x_k \}_{k=0}^{\infty}$ is a Riesz basis for $L^2({\mathbb{R}})$ with orthonormalizer $U$ then
for all $n$,
$$
\sum_{k=0}^n \left( \Delta^2(x_k) + \Delta^2(\widehat{x_k}) +
|\mu(x_k)|^2 + |\mu(\widehat{x_k})|^2 \right) \geq
\frac{(n+1)(2n+1)}{4\pi\norm{U}^2}.
$$
Thus, for every $A>0$, there are at most $8\pi A^2\norm{U}^2$ elements of the
basis $\{ x_k \}_{k=0}^{\infty}$
such that $|\mu(e_n)|$, $|\mu(\widehat{e_n})|$, $\Delta(e_n)$, $\Delta(\widehat{e_n})$
are all bounded by $A$. In particular,
$$
\max\{|\mu(x_k)|,\ |\mu(\widehat{x_k})|,\ \Delta(x_k),\ \Delta(\widehat{x_k}):\ 0 \leq k \leq n \}
\geq \frac{1}{\norm{U}}\sqrt{\frac{2n+1}{16\pi}}.
$$
\end{corollary}

\section{Finite dimensional approximations, spherical codes and the Umbrella Theorem} \label{sec3.ap}

\subsection{Spherical codes} \label{ap.sphcode.sec}

Let ${\mathbb{K}}$ be either ${\mathbb{R}}$ or ${\mathbb{C}}$, and let $d\geq1$ be a fixed integer. We equip ${\mathbb{K}}^d$ with the standard
Euclidean scalar product and norm. We denote by ${\mathbb{S}}_d$ the unit sphere of ${\mathbb{K}}^d$.

\begin{definition}
Let $A$ be a subset of $\{z\in{\mathbb{K}}\,:\ |z|\leq 1\}$. A spherical $A$-code is a finite subset $V \subset {\mathbb{S}}_d$
such that if $u,v \in V$ and $u\not=v$ then $\scal{u,v}\in A$.

\end{definition}

Let $N_{\mathbb{K}}(A,d)$ denote the maximal cardinality of a spherical $A$-code.
This notion has been introduced in \cite{JP.DeGoSi} in the case ${\mathbb{K}}={\mathbb{R}}$ where upper-bounds
on $N_{\mathbb{R}}(A,d)$ have been obtained.  These are important quantities in geometric combinatorics,
and there is a large associated literature.  Apart from \cite{JP.DeGoSi},
the results we use can all be found in \cite{JP.CoSl}.

Our prime interest is in the quantity
$$N_{\mathbb{K}}^s(\alpha,d)=\begin{cases}N_{\mathbb{R}}([-\alpha,\alpha],d),&\mbox{when }{\mathbb{K}}={\mathbb{R}}\\
N_{\mathbb{C}}(\{z\in {\mathbb{C}}:|z|\leq \alpha\},d),&\mbox{when }{\mathbb{K}}={\mathbb{C}}\\ \end{cases}$$ for $\alpha\in (0,1]$.
Of course $N_{\mathbb{R}}^s(\alpha,d)\leq N_{\mathbb{C}}^s(\alpha,d)$.
Using the standard identification of ${\mathbb{C}}^d$ with ${\mathbb{R}}^{2d}$, namely
identifying $Z=(x_1+iy_1,\ldots, x_d + iy_d)\in{\mathbb{C}}^d$ with
$\tilde Z=(x_1,y_1,\ldots, x_d, y_d)\in{\mathbb{R}}^{2d}$, we have
$\scal{\tilde Z,\tilde Z'}_{{\mathbb{R}}^{2d}}=\mbox{Re}\,\scal {Z,Z'}_{{\mathbb{C}}^d}$.
Thus $N_{\mathbb{C}}^s(\alpha,d)\leq N_{\mathbb{R}}^s(\alpha,2d)$.

In dimensions $d=1$ and $d=2$ one can compute the following values for $N_{\mathbb{K}}^s(\alpha,d)$:
\begin{itemize}
\item $N_{\mathbb{R}}^s(\alpha,1)=1$
\item If $0 \leq \alpha < 1/2$ then $N_{\mathbb{R}}^s(\alpha,2)=2$
\item If $\cos\frac{\pi}{N}\leq \alpha<\cos\frac{\pi}{N+1}$ and $3 \leq N$
then $N_{\mathbb{R}}^s(\alpha,2)=N$.
\end{itemize}
In higher dimensions, one has the following result.
\begin{lemma} \label{ap.sphcodebnd.lem}
If $0\leq \alpha<\frac{1}{d}$ then $N_{\mathbb{K}}^s(\alpha,d)=d$.
\end{lemma}
\begin{proof}
An orthonormal basis of ${\mathbb{K}}^d$
is a spherical $[-\alpha,\alpha]$-code so that $N_{\mathbb{K}}^s(\alpha,d)\geq d$.
For the converse, let $\alpha<1/d$ and assume towards a contradiction that
$w_0,\ldots,w_d$ is a spherical $[-\alpha,\alpha]$-code.
Indeed, let us show that $w_0,\ldots,w_d$ would be linearly independent in ${\mathbb{K}}^d$.
Suppose that $\displaystyle\sum_{j=0}^{d}\lambda_jw_j=0,$ and without loss of generality
that $|\lambda_j|\leq|\lambda_0|$ for $j=1,\ldots,d$.
Then $\lambda_0\norm{w_0}^2=\displaystyle-\sum_{j=1}^{d}\lambda_j\scal{w_j,w_0}$
so that
$|\lambda_0|\leq |\lambda_0|d\alpha.$
As $d\alpha<1$ we get that $\lambda_0=0$ and then $\lambda_j=0$  for all $j$.
\end{proof}

In general, it is difficult to compute $N_{\mathbb{K}}^s(\alpha,k)$.  A coarse
estimate using volume counting proceeds as follows.
\begin{lemma} \label{ap.expbnd.lem}
If $0 \leq \alpha <1$ is fixed, then there exist constants
$0<a_1<a_2$ and $0<C$ such that for all $d$
$$
\frac{1}{C}e^{a_1 d}\leq N_{\mathbb{K}}^s(\alpha,d)\leq Ce^{a_2 d}.
$$
Moreover, for $\alpha\leq 1/2$ one has
$N_{\mathbb{K}}^s(\alpha,d)  \leq 3^d$ if ${\mathbb{K}}={\mathbb{R}}$, and $N_{\mathbb{K}}^s(\alpha,d) \leq 9^d$
if ${\mathbb{K}}={\mathbb{C}}$.
\end{lemma}

\begin{proof}
\noindent The counting argument for the upper bound proceeds as follows.
Let $\{w_j\}_{n=1}^N$ be a spherical $A$-code, with $A =[-\alpha, \alpha]$
or $A= \{z\in {\mathbb{C}} : |z| \leq \alpha \}.$
For $j\not=k$, one has
$$
\norm{w_j-w_k}^2=\norm{w_j}^2+\norm{w_k}^2+2\mbox{Re}\,\scal{w_j,w_k}
\geq 2-2\alpha.
$$
So, the open balls $\displaystyle B\left(w_j,\sqrt{\frac{1-\alpha}{2}}\right)$ of center $w_j$ and radius
$\displaystyle\sqrt{\frac{1-\alpha}{2}}$ are all disjoint and included in the ball of center $0$ and radius
$\dst1+\sqrt{\frac{1-\alpha}{2}}$. Therefore
$$
Nc_d\left(\frac{1-\alpha}{2}\right)^{hd/2}\leq c_d\left(1+\sqrt{\frac{1-\alpha}{2}}\right)^{hd}
$$
where $c_d$ is the volume of the unit ball in ${\mathbb{K}}^d$,
$h=1$ if ${\mathbb{K}}={\mathbb{R}}$ and $h=2$ if ${\mathbb{K}}={\mathbb{C}}$.
This gives the bound $N\leq \left(1+\sqrt{\frac{2}{1-\alpha}}\right)^{hd}$.
Note that for $\alpha\leq 1/2$ we get $N\leq 3^d$ if ${\mathbb{K}}={\mathbb{R}}$ and $N\leq 9^d$ if ${\mathbb{K}}={\mathbb{C}}$.
The lower bound too may be obtained by a volume counting argument, see \cite{JP.CoSl}.
\end{proof}

The work of Delsarte, Goethals, Seidel, \cite{JP.DeGoSi} provides a method for
obtaining more refined estimates on the size of spherical codes.
For example, taking $\beta = - \alpha$ in Example 4.5 of \cite{JP.DeGoSi}
shows that
if $\alpha<\frac{1}{\sqrt{d}}$ then
\begin{equation} \label{delsarte_bnd.ap}
\displaystyle N_{\mathbb{R}}^s(\alpha,d)\leq \frac{(1-\alpha^2)d}{1-\alpha^2d}.
\end{equation}
Equality can only occur for spherical $\{-\alpha,\alpha\}$-codes.
Also, note that if $\alpha=\frac{1}{\sqrt{d}}\sqrt{1-\frac{1}{d^k}}$, then
$\frac{1-\alpha^2}{1-\alpha^2d}d\sim d^{k+1}$.

\subsection{Approximations of orthonormal bases} \label{ap.apponb.sec}

We now make a connection between the cardinality of spherical codes and
projections of orthonormal bases.

Let $\mathcal{H}$ be a Hilbert space over ${\mathbb{K}}$ and let $\Psi=\{\psi_k\}_{k=1}^{\infty}$ be an orthonormal
basis for $\mathcal{H}$. For an integer $d\geq 1$, let ${\mathbb{P}}_d$ be the orthogonal
projection on the span of $\{\psi_1,\ldots,\psi_d\}$.
For $\varepsilon>0$, we say that an element $f\in \mathcal{H}$
is $\varepsilon,d$-approximable
if $\norm{f-{\mathbb{P}}_df}_{\mathcal{H}}<\varepsilon$, and define
${{\mathcal S}}_{\varepsilon,d}$ to be the set of
all of $f\in {\mathcal{H}}$ with
$||f||_{\mathcal{H}}=1$ that are $\varepsilon,d$-approximable.
We denote by $A_{\mathbb{K}}(\varepsilon,d)$ the maximal cardinality of an orthonormal sequence
in ${{\mathcal S}}_{\varepsilon,d}$.

\begin{examplenum}
Let $\{e_j\}_{j=1}^n$ be the canonical basis for ${\mathbb{R}}^n$, and let
$\{\psi_j\}_{j=1}^{n-1}$ be
an orthonormal basis for $V^{\perp}$,
where $V = {\rm span} \{(1,1,\ldots,1)\}$.
Then $\norm{e_k-P_{n-1}e_k}_2=\frac{1}{\sqrt{n}}$ holds for each
$1 \leq k \leq n$,
and hence $A_{\mathbb{R}}(\frac{1}{\sqrt{n}},n-1)\geq n$.
\end{examplenum}

Our interest in spherical codes stems from the following result, cf. \cite[Corollary 1]{JP.Po}.

\begin{proposition}
\label{JP.prop:estimate}
If $0<\varepsilon<1/\sqrt{2}$ and  $\alpha=\frac{\varepsilon^2}{1-\varepsilon^2},$
then $A_{\mathbb{K}}(\varepsilon,d)\leq N_{\mathbb{K}}^s(\alpha,d)$.
\end{proposition}

\begin{proof}
Let $\{\psi_k\}_{k=1}^{\infty}$ be an orthonormal basis for $\mathcal{H}$,
and let ${\mathcal S}_{\varepsilon,d}$ and ${\mathbb{P}}_d$ be as above.
Let $\{f_j\}_{j=1}^N \subset {\mathcal{H}}$
be an orthonormal set contained in ${\mathcal S}_{\varepsilon,d}$.
For each $k=1,\ldots,N$, $j=1,\ldots,d$,
let $a_{k,j}=\scal{f_k,\psi_j}$ and write ${\mathbb{P}}_df_k:=\displaystyle\sum_{j=1}^da_{k,j}\psi_j$
so that $\norm{f_k-{\mathbb{P}}_df_k}_{\mathcal{H}}<\varepsilon$.

Write $v_k=(a_{k,1},\ldots,a_{k,d})\in{\mathbb{K}}^d$ then, for $k\not=l$
\begin{eqnarray}
\scal{v_k,v_l}&=&\scal{{\mathbb{P}}_df_k,{\mathbb{P}}_df_l}=\scal{{\mathbb{P}}_df_k-f_k+f_k,{\mathbb{P}}_df_l-f_l+f_l}\nonumber\\
&=&\scal{{\mathbb{P}}_df_k-f_k,{\mathbb{P}}_df_l-f_l}+\scal{{\mathbb{P}}_df_k-f_k,f_l}+\scal{f_k,{\mathbb{P}}_df_l-f_l}\nonumber\\
&=&\scal{{\mathbb{P}}_df_k-f_k,{\mathbb{P}}_df_l-f_l}+\scal{{\mathbb{P}}_df_k-f_k,f_l-{\mathbb{P}}_df_l}+\scal{f_k-{\mathbb{P}}_df_k,{\mathbb{P}}_df_l-f_l}\nonumber\\
&=&\scal{f_k-{\mathbb{P}}_df_k,{\mathbb{P}}_df_l-f_l}
\label{JP.eq:orsp}
\end{eqnarray}
since ${\mathbb{P}}_df_k-f_k$ is orthogonal to ${\mathbb{P}}_df_l$.
It follows from the Cauchy-Schwarz inequality
that $\abs{\scal{v_k,v_l}}\leq \varepsilon^2$.

On the other hand,
$$
\norm{v_k}_{{\mathbb{K}}^d}=\norm{{\mathbb{P}}_df_k}_{\mathcal{H}}
=(\norm{f_k}_{\mathcal{H}}^2-
\norm{f_k-{\mathbb{P}}_df_k}_{\mathcal{H}}^2)^{1/2}\geq(1-\varepsilon^2)^{1/2}.
$$
It follows that $w_k=\displaystyle\frac{v_k}{\norm{v_k}_{{\mathbb{K}}^d}}$
satisfies, for $k\not=l$,
$\abs{\scal{w_k,w_l}}=
\frac{\abs{\scal{v_k,v_l}}}{\norm{v_k}_{{\mathbb{K}}^d}\norm{v_l}_{{\mathbb{K}}^d}}\leq\frac{\varepsilon^2}{1-\varepsilon^2},$
and $\{w_k\}_{k=1}^N$ is a spherical $[-\alpha,\alpha]$-code in ${\mathbb{K}}^d$.
\end{proof}

Note that the proof only uses orthogonality in a mild way.
Namely, if instead
$\{f_j\}_{j=1}^N \subset \mathcal{H}$ with $\norm{f_j}_{\mathcal{H}} =1$
satisfies
$|\scal{f_j,f_k}| \leq \eta^2$ for $j\neq k$, then
Equation (\ref{JP.eq:orsp}) becomes
$\scal{v_k,v_l}=\scal{f_k-{\mathbb{P}}_df_k,{\mathbb{P}}_df_l-f_l}
+\scal{f_k,f_l},$ so that
$|\scal{v_k,v_l}| \leq \varepsilon^2+\eta^2$,
and the end of the proof shows that
$\displaystyle N\leq N_{\mathbb{K}}^s\left(\frac{\varepsilon^2+\eta^2}{1-\varepsilon^2},d\right)$.

In view of Proposition \ref{JP.prop:estimate},
it is natural ask the following question.
Given $\alpha=\frac{\varepsilon^2}{1-\varepsilon^2}$, is there a converse inequality
of the form $N_{\mathbb{K}}^s(\alpha,d)\leq CA_{\mathbb{K}}(\varepsilon',d')$
with $C>0$ an absolute constant and $\varepsilon\leq \varepsilon'\leq C\varepsilon$, $d\leq d'\leq Cd$?  Note that for $\varepsilon$ such that $\alpha<1/d$, we have
$A_{\mathbb{K}}(\varepsilon,d)=N_{\mathbb{K}}^s(\alpha,d)=d$.

\subsection{Prolate sphero\"\i dal wave functions} \label{ap.psw.sec}
In order to obtain quantitative versions of Sha\-pi\-ro's theorems, we will
make use of the prolate sphero\"\i dal wave functions.  For a detailed presentation on
prolate sphero\"\i dal wave functions see \cite{JP.SP,JP.LP1,JP.LP2}.

Fix $T,\Omega>0$ and let $\{\psi_n\}_{n=0}^{\infty}$ be the associated prolate spheroidal wave
functions, as defined in \cite{JP.SP}.  $\{\psi_n\}_{n=0}^{\infty}$ is an orthonormal basis for
$PW_\Omega\equiv
\{f\in L^2({\mathbb{R}})\,:\ \mbox{supp}\,\widehat{f}\subseteq [-\Omega,\Omega]\},$
and the $\psi_n$
are eigenfunctions of the differential operator
$$L=(T^2-x^2) \frac{{d^2}}{{d} x^2}
-2x \frac{{d}}{{d}x} - \frac{\Omega^2}{T^2}x^2.$$
As in the previous section, for an integer $d\geq 0$, define ${\mathbb{P}}_d$ to be the
projection onto
the span of $\psi_0,\ldots,\psi_{d-1},$ and for $\varepsilon>0$ define
$$
\mathcal{S}_{\varepsilon,d}=\{f\in L^2({\mathbb{R}})\,: \norm{f}_2=1, \norm{f-{\mathbb{P}}_df}_2 <\varepsilon\}.
$$
For the remainder of the paper, the
orthonormal basis used in the definitions of
$\mathcal{S}_{\varepsilon,d}$, ${\mathbb{P}}_d$, and
${A}_{\mathbb{K}} (\varepsilon, d)$, will always be chosen as the prolate
sphero\"\i dal wavefunctions.  Note that these quantities depend on
the choice of $T,\Omega$.

Finally, let
$$
{\mathcal P}_{T,\Omega,\varepsilon}=\left\{f\in L^2({\mathbb{R}})\,:\ \int_{|t|>T}|f(t)|^2\,\mbox{d}t\leq\varepsilon^2
\quad\mbox{and}\quad\int_{|\xi|>\Omega}|\widehat{f}(\xi)|^2\,\mbox{d}\xi\leq\varepsilon^2\right\}
$$
and ${\mathcal P}_{T,\varepsilon}={\mathcal P}_{T,T,\varepsilon}$.

\begin{theorem}[Landau-Pollak \cite{JP.LP2}]
\label{JP.th:LPth12}
Let $T,\ \varepsilon$ be positive constants and let $d=\lfloor 4T\Omega\rfloor+1$. Then,
for every $f\in {\mathcal P}_{T,\Omega,\varepsilon}$,
$$
\norm{f-{\mathbb{P}}_df}_2^2\leq 49\varepsilon^2\norm{f}_2^2.
$$
In other words, ${\mathcal P}_{T,\Omega,\varepsilon}\cap\{f\in L^2 ({\mathbb{R}})\,:\ \norm{f}_2=1\}\subset\mathcal{S}_{7\varepsilon,d}$.
\end{theorem}

It follows that the first $d=4T^2+1$  elements
of the prolate sphero\"\i dal basis well approximate
${\mathcal P}_{T,\varepsilon}$, and that ${\mathcal P}_{T,\varepsilon}$ is ``essentially'' $d$-dimensional.

\subsection{Generalized means and dispersions} \label{gen.md.ap}
As an application of the results on prolate spheroidal wavefunctions
and spherical codes, we shall address a more general version of the
mean-dispersion theorem.

Consider the following generalized means and variances.
For $p>1$ and $f\in L^2({\mathbb{R}})$ with $\norm{f}_2=1$, we define the following associated
\emph{$p$-variance}
$$
\Delta_p^2(f)=\inf_{a\in{\mathbb{R}}}\int|t-a|^p|f(t)|^2\mbox{d}t.
$$
One can show that the infimum is actually a minimum and is attained
for a unique $a\in{\mathbb{R}}$ that we call the \emph{$p$-mean}
$$
\mu_p(f)= \mbox{arg min}_{a \in {\mathbb{R}}}\,\int|t-a|^p|f(t)|^2\mbox{d}t.
$$
As before, define the \emph{$p$-dispersion} $\Delta_p(f)\equiv\sqrt{\Delta_p^2(f)}$.

The proof of the Mean-Dispersion Theorem for $p=2$ via the Rayleigh-Ritz technique
relied on the special relation \eqref{JP.eq:fundherm}
of means and dispersions with the Hermite operator.   In general, beyond the case $p=2$,
such simple relations are not present and the techniques of
Section \ref{sec2.ap} are not so easily applicable.
However, we shall show how to use
the combinatorial techniques from the beginning of this section
to obtain a quantitative version of Theorem \ref{ap.sh-meandisp} for generalized means and dispersions.

The following lemma is a modification of \cite[Lemma 1]{JP.Po}.
\begin{lemma}
\label{JP.lem:Po-prolate}
Let $A>0$ and $p>1$. Suppose $g\in L^2({\mathbb{R}})$, $\norm{g}_2=1$ satisfies
$$
|\mu_p(g)|,\ |\mu_p(\widehat{g})|,\ \Delta_p(g),\ \Delta_p(\widehat{g})\leq A.
$$
Fix $\varepsilon>0$, then $g\in{\mathcal P}_{A+(A/\varepsilon)^{2/p},\varepsilon}$.
\end{lemma}

This gives a simple proof of a strengthened version of Shapiro's Mean-Dispersion Theorem:
\begin{corollary}
\label{JP.cor:cor3}
Let $0<A$, $1<p<\infty,$ $0<\varepsilon <1/7\sqrt{2}$, $\alpha=49\varepsilon^2/(1-49\varepsilon^2)$,
and set
$d=\lfloor 4\bigl(A+(A/\varepsilon)^{2/p}\bigr)^2\rfloor+1$.

If $\{e_k\}_{k=1}^{N} \subset L^2 ({\mathbb{R}})$ is an orthonormal set such that
for all $1 \leq k \leq N$,
$$
|\mu_p(e_k)|,\ |\mu_p(\widehat{e_k})|,\
\Delta_p(e_k),\ \Delta_p(\widehat{e_k})\leq A,
$$
then $$N \leq N_{\mathbb{C}}^s(\alpha,d)\leq N_{\mathbb{R}}^s(\alpha,2d).$$
\end{corollary}

\begin{proof}
According to Lemma \ref{JP.lem:Po-prolate}, $e_1,\ldots,e_n$
are in ${\mathcal P}_{A+(A/\varepsilon)^{2/p},\varepsilon}$.
The definition of $d$ and Theorem \ref{JP.th:LPth12} show that
$\{ e_j \}_{j=1}^N \subset {\mathcal S}_{7\varepsilon,d}$.
According to Proposition \ref{JP.prop:estimate},
$N\leq A_{\mathbb{C}}(7\varepsilon,d)\leq N_{\mathbb{C}}^s(\alpha,d)\leq N_{\mathbb{R}}^s(\alpha,2d)$, where $\alpha=49\varepsilon^2/(1-49\varepsilon^2)$.
\end{proof}

This approach does not, in general, give sharp results.  For example,
in the case $p=2$ the bound obtained by Corollary \ref{JP.cor:cor3} is not as good as the one
given in Section 2.  To see this, assume that $p=2$ and $A\geq 1$.
Then $4A^2(1+1/\varepsilon)^2\leq d\leq 5A^2(1+1/\varepsilon)^2$.
In order to apply the Delsarte, Goethals, Seidal bound (\ref{delsarte_bnd.ap})
we will now chose
$\varepsilon$ so that $\alpha<\frac{1}{2\sqrt{d}}$ which will then give that $n\leq 4d$.
Our aim is thus to take $d$ is as small as possible by chosing
$\varepsilon$ as large as possible.

For this, let us first take $\varepsilon\leq 1/50$ so that $\alpha\leq 50\varepsilon^2$.
It is then enough that $50\varepsilon^2\leq\frac{1}{4A(1+1/\varepsilon)}$, that is
$\varepsilon^2+\varepsilon-\frac{1}{200A}\leq 0$. We may thus take
$\varepsilon= \frac{\sqrt{1+\frac{1}{50A}}-1}{2}$. Note that, as $A\geq 1$,
we get that $\varepsilon\leq\frac{\sqrt{1+\frac{1}{50}}-1}{2}<1/50$.
This then gives
$$
n\leq 20d\leq 20A^2\left(1+\frac{2}{\sqrt{1+\frac{1}{50A}}-1}\right)^2
=20A^2\left(1+100A\Bigl(\sqrt{1+\frac{1}{50A}}+1\Bigr)\right)^2
\leq C A^4.
$$
In particular, the combinatorial methods allow one to take $N= CA^4$ in Corollary
\ref{JP.cor:cor3}, whereas the sharp methods of Section \ref{ap.sharp.md.sec}
give $N = 8 \pi A^2,$ see Corollary \ref{ap.cor.opt.md}.

\subsection{The Quantitative Umbrella Theorem}
\label{JP.sec:QUT}
A second application of our method is a quantitative form of Shapiro's umbrella theorem.
As with the mean-dispersion theorem, Shapiro's proof does not provide a bound on the number
of elements in the sequence.  As before, the combinatorial approach is well suited to this setting
whereas the approach of Section \ref{sec2.ap} is not easily applicable.

Given $f \in L^2 (\mathbb{R})$ and $\varepsilon>0$, define
$$
C_f(\varepsilon)=\inf\left\{ T \geq 0 \,:\ \int_{|t|>T}| f(t)|^2\leq\varepsilon^2\norm{f}_2^2
\right\}.$$  Note that if $f$ is not identically zero then
for all $0<\varepsilon<1$ one
has $0 < C_f (\varepsilon) < \infty$.

\begin{theorem}
\label{JP.th:umbrella}
Let $\varphi,\psi\in L^2({\mathbb{R}})$ and
$M = \min \{ \norm{\varphi}_2,\norm{\psi}_2 \} \geq 1$.
Fix $\frac{1}{50M} \geq \varepsilon >0$,
$T >\max \{ C_\varphi(\varepsilon),C_\psi(\varepsilon) \}$,
and $d = \lfloor 4T^2\rfloor+1$.

If $\{e_n\}_{n=1}^N$ is an orthonormal sequence in $L^2({\mathbb{R}})$
such that for all $1 \leq n \leq N,$ and for almost all $x, \xi \in{\mathbb{R}}$,
\begin{equation}
\label{JP.eq:umbrella}
|e_n(x)|\leq|\varphi(x)|\quad\mbox{and}\quad|\widehat{e}_n(\xi)|\leq|\psi(\xi)|,
\end{equation}
then
\begin{equation} \label{ap.bnd.umbrella.thm}
N
\leq N^s_{\mathbb{C}}(50\varepsilon^2M^2,d) \leq N^s_{\mathbb{R}}(50\varepsilon^2M^2,2d).
\end{equation}

In particular, $N$
is bounded by an absolute constant depending
only on $\varphi$ and $\psi$.
\end{theorem}

\begin{proof}
By (\ref{JP.eq:umbrella}), $T>\max \{ C_\varphi(\varepsilon),C_\psi(\varepsilon) \}$,
implies $\{ e_n \}_{n=1}^N \subset{\mathcal P}_{T,\varepsilon M}$.
According to Theorem \ref{JP.th:LPth12},
${\mathcal P}_{T,\varepsilon M}\subset {\mathcal S}_{7\varepsilon M,d}$.
It now follows from Proposition \ref{JP.prop:estimate}, that
$$
N \leq \mathbb{A}_{{\mathbb{C}}} \left( 7 \varepsilon M, d \right)
\leq N^s_{\mathbb{C}} \left( \frac{49\varepsilon^2M^2}{1 - 49 \varepsilon^2 M^2},d \right)
\leq N^s_{\mathbb{C}}(50\varepsilon^2M^2,d) \leq N^s_{\mathbb{R}}(50\varepsilon^2M^2,2d)
.
$$
\end{proof}

Let us give two applications where one may get an explicit upper bound
by making a proper choice of $\varepsilon$ in the proof above.

\begin{proposition}
\label{JP.ex:power}
Let $1/2<p$ and
$\sqrt{\frac{2p-1}{2}} \leq C$
be fixed.
If $\{e_n \}_{n=1}^N \subset L^2({\mathbb{R}})$ is an orthonormal set such that
for all $1 \leq n \leq N$, and for almost every $x,\xi \in {\mathbb{R}}$,
$$|e_n(x)|\leq\frac{C}{(1+|x|)^p} \ \ \ \hbox{ and } \ \ \
|\widehat{e}_n(\xi)|\leq\frac{C}{(1+|\xi|)^p},$$
then
$$
N\leq \begin{cases}
9^{\left(\frac{200\sqrt{2}C}{\sqrt{2p-1}}\right)^{\frac{4}{2p-1}}}
&\mbox{ if } 1/2 < p,\\
16\left(\frac{400C^2}{2p-1}\right)^{\frac{1}{p-1}}&\mbox{ if }
1 < p \leq 3/2,\\
{4}
\left(\frac{500C^2}{2p-1}\right)^{\frac{2}{2p-3}}&\mbox{ if } 3/2 < p.\\
\end{cases}$$

\end{proposition}

\begin{proof}
If $\varphi(x)=\frac{C}{(1+|x|)^p}$, then
$M = ||\varphi||_2 = C\sqrt{\frac{2}{2p-1}} \geq 1$,
and a computation for $0 < \varepsilon \leq 1$ shows that
$C_\varphi(\varepsilon)=\frac{1}{\varepsilon^{2/(2p-1)}}-1.$
Let $\delta =\delta(\varepsilon)=\frac{4}{\varepsilon^{4/(2p-1)}}$ and
$\alpha = \alpha(\varepsilon) =\frac{100 C^2\varepsilon^2}{2p-1}$.
Taking $T = C_{\varphi}(\varepsilon)$ implies that
$d = \lfloor 4T^2\rfloor+1
\leq \delta(\varepsilon)$.

If $0 < \varepsilon \leq \frac{1}{50M}$, then
Theorem \ref{JP.th:umbrella} gives the bound
$N\leq N^s_{\mathbb{C}}\left(\alpha (\varepsilon ),\delta (\varepsilon ) \right)$.
We shall chose $\varepsilon$ differently for the various cases.

{\em Case 1.}  For the case $1/2 <p,$ take
$\varepsilon=\frac{1}{50M},$
and use the exponential bound given by
Lemma \ref{ap.expbnd.lem} for $N^s_{\mathbb{C}}(\alpha (\varepsilon), \delta (\varepsilon))$
to obtain the desired estimate.

{\em Case 2.}  For the case  $1<p\leq 3/2$,
let $\varepsilon_0=
\left(\frac{\sqrt{2p-1}}{20C}\right)^{\frac{2p-1}{2(p-1)}}$,
$\alpha=\alpha(\varepsilon_0)$, and $\delta=\delta(\varepsilon_0)$.
Note that $\alpha = \frac{1}{2\sqrt{\delta}}< \frac{1}{\sqrt{2\delta}}$,
and also
that $\varepsilon_0 \leq \frac{1}{50M},$ since $1 < p \leq 3/2$.
Thus the bound (\ref{delsarte_bnd.ap}) yields
$N \leq N^s_{\mathbb{R}}(\alpha,2\delta)= 4(1-\alpha^2)\delta\leq 4\delta$.
The desired estimate follows.

{\em Case 3.}  For the case $3/2<p$, define
$\varepsilon_1=\left(\frac{\sqrt{2p-1}}{50C\sqrt{2}}\right)^{\frac{2p-1}{2p-3}}$
and note that  $\varepsilon_1 \leq \frac{1}{50M}.$
Since $3/2<p$, taking
$\varepsilon<\varepsilon_1$, $\alpha=\alpha(\varepsilon)$, $\delta=\delta(\varepsilon),$ implies
that $\alpha(\varepsilon) < 1/ \delta(\varepsilon)$.
Thus, by Lemma \ref{ap.sphcodebnd.lem}, $N \leq \delta(\varepsilon)$
for all $\varepsilon < \varepsilon_1$.  Hence,
$N \leq \delta (\varepsilon_1),$ and the desired estimate follows.

\end{proof}

Note that in the case $1/2 <p$, the upper bound
in Proposition \ref{JP.ex:power} approaches infinity
as $p$ approaches $1/2$.  Indeed, we refer the reader to
the counterexamples for $p<1/2$ in \cite{JP.DeR, By.ap}.
The case $p=1/2$ seems to be open as \cite{JP.DeR} need an extra logarithmic factor
in their construction. For perspective in the case $3/2<p$,
if one takes $C = C_p = \sqrt{\frac{2p -1}{2}}$,
then the upper bound in Proposition \ref{JP.ex:power}
approaches $4$ as $p$ approaches infinity.

\begin{proposition}
Let $0<a\leq 1$ and $(2a)^{1/4} \leq C$ be fixed.
If $\{e_n\}_{n=0}^N \subset L^2 ({\mathbb{R}})$ is an orthonormal set such that
for all $n$ and for almost every $x, \xi \in {\mathbb{R}}$
$$|e_n(x)|\leq Ce^{-\pi a|x|^2} \ \ \ \hbox{ and } \ \ \
|\widehat{e}_n(\xi)|\leq Ce^{-\pi a|\xi|^2},$$
then
$$N \leq  2+ \frac{8}{a\pi} \max \Big\{
2\ln \left( \frac{50 C \sqrt{\pi} e^{\pi}}{a^{1/4}} \right),
\ln \left( \frac{50 \pi C^2 e^{\pi a /2}}{a^{5/2}e^{2\pi}} \right)
\Big\}.$$
\end{proposition}

\begin{proof}  Let $\gamma_a(x)=Ce^{-\pi a|\xi|^2}$ and let
$C_a(\varepsilon)=C_{\gamma_a}(\varepsilon)$.
First note that
\begin{eqnarray*}
\int_{|t|>T} |\gamma_a (t)|^2 dt &=&
\int_{|t|>T}C^2e^{-2\pi a|t|^2} {d}t
=\frac{2C^2}{\sqrt{a}}\int_{T\sqrt{a}}^{\infty}
\frac{(1+s^2)e^{-2\pi s^2}}{1 +s^2} {d}s\\
&\leq&
\frac{C^2\pi (1+aT^2)}{\sqrt{a}} e^{-2\pi a T^2},
\end{eqnarray*}
while $M=||\gamma_a||_2 =
(\int_{{\mathbb{R}}} C^2e^{-2\pi a|t|^2} {d}t)^{1/2}
=\frac{C}{(2a)^{1/4}}$. In particular, $\norm{\gamma_a}_2\geq1$.
Now for every $T>0$, set
$\varepsilon(T)=2^{-1/4} \sqrt{\pi} \sqrt{1+aT^2}e^{-\pi a T^2}$, so that
$C_a\bigl(\varepsilon(T)\bigr) \leq T$.

By Theorem \ref{JP.th:umbrella}, we get that
$N\leq N_{\mathbb{R}}^s(50\varepsilon^2(T)M^2,8T^2+2)$, provided $\varepsilon(T) \leq \frac{1}{50M}$.
Let us first see what condition should be imposed on $T$ to have
$\varepsilon(T) \leq \frac{1}{50M}$.  Setting $s=(1+aT^2)$, this condition is
equivalent to $\sqrt{s} e^{- \pi s} \leq
\frac{e^{-\pi} a^{1/4}}{50 C\sqrt{\pi}}.$
Thus, it suffices to take $s \geq
\frac{2}{\pi} \ln \left( \frac{50 C \sqrt{\pi} e^{\pi}}{a^{1/4}} \right)$,
and
$T^2 \geq \frac{2}{a\pi}
\ln \left( \frac{50 C \sqrt{\pi} e^{\pi}}{a^{1/4}} \right)$.

We will now further choose $T$ large enough to have $50\varepsilon(T)^2M^2<
\frac{1}{8T^2+2}$,
so that Lemma \ref{ap.sphcodebnd.lem} will imply
$N \leq N_{\mathbb{R}}^s(50\varepsilon(T)^2M^2,8T^2+2)=8T^2+2$.
This time, the condition reads
$(1 + aT^2)(1 +4T^2) e^{-2\pi a T^2} < \frac{\sqrt{a}}{50 \pi C^2}$.
Let $r= a(4T^2 +1)$.  Thus, it suffices to take
$r^2 e^{-\frac{\pi}{2} r} <
\frac{a^{5/2} e^{2\pi}}{50 \pi C^2 e^{{\pi a}/{2}}}$
It is enough to take
$r> \frac{4}{\pi} \ln \left( \frac{50 \pi C^2
e^{\pi a /2}}{a^{5/2}e^{2\pi}} \right)$,
and $T^2 > \frac{1}{a\pi}
\ln \left( \frac{50 \pi C^2 e^{\pi a /2}}{a^{5/2}e^{2\pi}} \right)$.

Combining the bounds for $T^2$ from the previous two paragraphs
yields
$$N \leq  2+ \frac{8}{a\pi} \max \Big\{
2\ln \left( \frac{50 C \sqrt{\pi} e^{\pi}}{a^{1/4}} \right),
\ln \left( \frac{50 \pi C^2 e^{\pi a /2}}{a^{5/2}e^{2\pi}} \right)
\Big\}.$$

\end{proof}

A careful reading of the proof of the Umbrella Theorem shows the following:
\begin{proposition}
Let $0<C,$ and let $1\leq p,q,\widehat{p},\widehat{q}\leq \infty$
satisfy $\displaystyle\frac{1}{p}+\frac{1}{q}=1$ and
$\displaystyle\frac{1}{\widehat{p}}+\frac{1}{\widehat{q}}=1$. Let $\varphi\in L^{2p}({\mathbb{R}})$ and $\psi\in L^{2\widehat{p}}({\mathbb{R}})$, and suppose that
$\varphi_k\in L^{2q}({\mathbb{R}})$ and $\psi_k\in L^{2\widehat{q}}({\mathbb{R}})$
satisfy $\norm{\varphi_k}_{2q} \leq C$, $\norm{\psi_k}_{2\widehat{q}}\leq C$.
There exists $N$ such that, if $\{e_k\}\subset L^2 ({\mathbb{R}})$
is an orthonormal set which for all $k$ and
almost every $x, \xi \in{\mathbb{R}}$ satisfies
$$
|e_k(x)|\leq \varphi_k(x) \ \varphi(x) \quad \ \ \ \hbox{ and } \ \ \
\quad |\widehat{e_k} (\xi)|\leq \psi_k(\xi) \ \psi(\xi),
$$
then $\{e_k\}$ has at most $N$ elements.  As with previous results,
a bound for $N$ can be obtained in terms of spherical codes.  The
bound for $N$ depends only on $\varphi, \psi, C$.
\end{proposition}

Indeed, let $\varepsilon>0$ and take $T>0$ big enough to have $\displaystyle\int_{|t|>T}|f(t)|^{2p}\mbox{d}t\leq\varepsilon^p/C^{2/p}$. Then
\begin{eqnarray*}
\int_{|t|>T}|e_k(t)|^2\,\mbox{d}t
&\leq&\int_{|t|>T}|\varphi_k(t)\varphi(t)|^2\,\mbox{d}t
\leq \left(\int_{|t|>T}|\varphi_k(t)|^{2q}\,\mbox{d}t\right)^{1/q}\left(\int_{|t|>T}|\varphi(t)|^{2p}\,\mbox{d}t\right)^{1/p}\\
&\leq& C^2(\varepsilon^p/C^{2/p})^{1/p}=\varepsilon.
\end{eqnarray*}
A similar estimate holds for $\widehat{e_k}$ and we conclude as in the proof of the Umbrella Theorem.

\subsection{Angles in Riesz bases} \label{ap.rieszang.sec}

Let us now conclude this section with a few remarks on Riesz bases.
Let $\{x_k\}_{k=0}^{\infty}$  be a Riesz basis for $L^2({\mathbb{R}})$ with orthogonalizer $U$
and recall that, for every sequence $\{a_n\}_{n=0}^{\infty} \in\ell^2$,
\begin{equation}
\frac{1}{\norm{U}^2}\sum_{n=0}^{\infty}|a_n|^2
\leq \norm{\sum_{n= 0}^{\infty} a_nx_n}_2^2\leq\norm{U^{-1}}^2\sum_{n=0}^{\infty} |a_n|^2.
\label{JP.eq:riesz2}
\end{equation}

Taking $a_n=\delta_{n,k}$ in (\ref{JP.eq:riesz2}) shows that
$\frac{1}{\norm{U}}\leq\norm{x_k}_2 \leq\norm{U^{-1}}$.
Then taking $a_n=\delta_{n,k}+\lambda\delta_{n,l}$, $k\not=l$ and $\lambda=t,-t$, $t>0$ gives
$$
\frac{1}{\norm{U}^2}(1+t^2)\leq\norm{x_k}_2^2+t^2\norm{x_l}_2^2+2t\abs{\mbox{Re}\scal{x_k,x_l}}
\leq\norm{U^{-1}}^2(1+t^2)
$$
thus $\abs{\mbox{Re}\scal{x_k,x_l}}^2$ is
\begin{eqnarray*}
&\leq&\min\Bigl((\norm{x_k}_2^2-\norm{U}^{-2})(\norm{x_l}_2^2-\norm{U}^{-2}),
(\norm{U^{-1}}^2-\norm{x_k}_2^2)(\norm{U^{-1}}^2-\norm{x_l}_2^2)\Bigr)\\
&\leq&\norm{x_k}_2^2\norm{x_l}_2^2\min\ent{
\left(1-\frac{1}{\norm{U}^2\norm{U^{-1}}^2}\right)^2
,\left(\frac{\norm{U^{-1}}^2}{\norm{U}^2}-1\right)^2}
\end{eqnarray*}
while taking $\lambda=it,-it$, $t>0$ gives the same bound for $\abs{\mbox{Im}\scal{x_k,x_l}}^2$.
It follows that
\begin{equation}
\label{JP.eq:c(u)}
\abs{\scal{x_k,x_l}}\leq C(U)\norm{x_k}_2 \norm{x_l}_2 \leq C(U)\norm{U^{-1}}^2
\end{equation}
where
$$
C(U):=\sqrt{2}
\min\ent{1-\left(\frac{1}{\norm{U}\norm{U^{-1}}}\right)^2,\left(\frac{\norm{U^{-1}}}{\norm{U}}\right)^2-1}.
$$
We may now adapt the proof of Proposition \ref{JP.prop:estimate} to Riesz basis:

\begin{proposition}
\label{JP.prop:estimateriesz}
Let $\{\psi_k)_{k=1}^{\infty}$ be an orthonormal basis for $L^2({\mathbb{R}})$.
Fix $d\geq 0$ and let ${\mathbb{P}}_d$ be the projection on the span of $\{\psi_1,\ldots,\psi_{d-1}\}$.

Let $\{x_k\}_{k=1}^{\infty}$ be a Riesz basis for $L^2({\mathbb{R}})$ and let $U$ be its orthogonalizer. Let $\varepsilon>0$ be such that
$\varepsilon<\min\left(\norm{U}^{-2},\sqrt{\frac{\norm{U}^{-2}-C(U)\norm{U^{-1}}^2}{2}}\right)$ and let
\begin{equation}
\label{JP.eq:defalpha}
\alpha=\frac{\varepsilon^2+C(U)\norm{U^{-1}}^2}{\norm{U}^{-2}-\varepsilon^2}.
\end{equation}
If $\{x_k\}_{k=1}^N$ satisfies
$\norm{x_k-{\mathbb{P}}_dx_k}_2<\varepsilon$ then $N \leq N_{\mathbb{K}}^s(\alpha,d)$.
\end{proposition}

\begin{proof}
Assume without loss of generality that $x_0,\ldots,x_N$ satisfy $\norm{x_k-{\mathbb{P}}_dx_k}<\varepsilon$
and let $a_{k,j}=\scal{x_k,\psi_j}$.

Write $v_k=(a_{k,1},\ldots,a_{k,d})\in{\mathbb{K}}^d$ then, the same computation as in (\ref{JP.eq:orsp}), for $k\not=l$
$$
\scal{v_k,v_l}=\scal{x_k-{\mathbb{P}}_dx_k,{\mathbb{P}}_dx_l-x_l}+\scal{x_k,x_l}
$$
thus $\abs{\scal{v_k,v_l}}\leq\varepsilon^2+\abs{\scal{x_k,x_l}}$.
On the other hand
$$
\norm{v_k}=\norm{{\mathbb{P}}_dx_k}=(\norm{x_k}^2-\norm{x_k-{\mathbb{P}}_df_k}^2)^{1/2}\geq(\norm{U}^{-2}-\varepsilon^2)^{1/2}
$$
It follows from (\ref{JP.eq:c(u)}) that $w_k=\displaystyle\frac{v_k}{\norm{v_k}}$ satisfies, for $k\not=l$,
$$
\abs{\scal{w_k,w_l}}\leq\frac{\varepsilon^2+C(U)\norm{U^{-1}}^2}{\norm{U}^{-2}-\varepsilon^2}
$$
and $\{w_k\}$ forms a spherical $[-\alpha,\alpha]$-code in ${\mathbb{K}}^d$.
\end{proof}

Note that the condition on $\varepsilon$ implies that $0<\alpha<1$.
Also note that if $U$ is a near isometry in the sense that $(1+\beta)^{-1}\leq\norm{U}^2\leq\norm{U^{-1}}^2\leq 1+\beta$
then $C(U)\leq\sqrt{2}\frac{\beta(2+\beta)}{(1+\beta)^2}$ and
$\alpha\leq\frac{(1+\beta)\varepsilon^2+\beta(2+\beta)}{1-(1+\beta)\varepsilon^2}$. In particular, if $U$ is near enough to an isometry,
meaning that $\beta$ is small enough, then this $\alpha$ is comparable with the $\alpha$ of Proposition
\ref{JP.prop:estimate}.

As a consequence, we may then easily adapt the proof of results that relied on Proposition \ref{JP.prop:estimate}
to the statements about Riesz bases. For example, an Umbrella Theorem for Riesz bases
reads as follows:

\begin{theorem}
\label{JP.th:umbrellaR}
Let $\varphi,\psi\in L^2({\mathbb{R}})$ with $\norm{\varphi}_2,\norm{\psi}_2 \geq1$.
Let $\{f_n\}_{n=1}^{\infty}$
be a Riesz basis for $L^2({\mathbb{R}})$
with orthonormalizer $U$ that is near enough to an isometry
$(1+\beta)^{-1}\leq\norm{U}^2\leq\norm{U^{-1}}^2\leq 1+\beta$ with $\beta$ small enough.
Then there exists a constant $N=N(\varphi,\psi,\beta)$ depending only on $\varphi, \psi$ and
$\beta$, such that the number of terms of the basis that satisfies
$$|f_n(x)|\leq|\varphi(x)|\quad\mbox{and}\quad|\widehat{f}_n(\xi)|\leq|\psi(\xi)|$$
for almost all $x, \xi\in{\mathbb{R}}$ is bounded by $N$.  As with previous results,
a bound on $N$ can be given in terms of spherical codes.
\end{theorem}

\subsection*{Acknowledgements}  A portion of this work was performed during the Erwin Schr\"odinger
Institute (ESI) Special Semester on ``Modern methods of time-frequency analysis.''
The authors gratefully acknowledge ESI for its hospitality
and financial support.  The authors also thank
Professor H.S. Shapiro for valuable comments related to the material.