$\newcommand\norm[1]{\lVert#1\rVert}\newcommand\abs[1]{\lvert#1\rvert}$I'm reading [this article by Boyd and Chua [1]](https://web.stanford.edu/%7Eboyd/papers/pdf/fading_volterra.pdf), in which they prove the approximability of arbitrary time-invariant (TI) operators (or filters) with polynomials of some basis filters with separation and *fading memory* (FM) properties. This shows that those operators, indeed have fintie Volterra expansion approximates.


Consider the following:


* $C(\Bbb R)$ as the space of bounded continuous functions $u: \Bbb R \to \Bbb R$,
* $\norm u = \sup\_{t \in \Bbb R} |u(t)|$ being their norm,
* $\Bbb R\_-$ as $\{t : t \le 0 \}$, with analogous definitions for $C(\Bbb R\_-)$ and $\norm\cdot$ as above,
* and time-invariant functionals $F: C(\Bbb R\_-) \to \Bbb R$ and time-invariant operators $N: C(\Bbb R) \to C(\Bbb R)$.


* $K=\bigl\{u\in C(\Bbb R): \abs{u(t)}\le M\_1\land\abs{u(s)-u(t)}\le M\_2(s-t)\;\forall t, s\in\Bbb R, t<s\bigr\}$: explicitly, $K$ is the space of bounded uniformly Lipschitz continuous functions. Functions belonging to $K$ are called *signals*.


* $K\_-=\{u\in K : u(t)=0\text{ if }t>0\}$: Boyd & Chua prefer to define $K\_-$ by using a "projection" operator $P$ such that
$$\DeclareMathOperator{\dmu}{d\!}
Pu(t) =
\begin{cases}
u(t) & t\le 0\\
0 & t>0
\end{cases}
$$ and then noting that $K\_- =P K$. Be it noted that $K\_-$ is compact in $C(\Bbb R)$ but only with respect to the weighted $\sup$ norm
defined as
$$
\norm u\_w=\sup\_{t\le 1} \abs{u(t)w(-t)}
$$
where $w:\Bbb R\_+\to(0,1]$ is a weight function such that $\lim\_{t\to\infty} w(t)=0$ (see below).
* For every TI operator $N$, an associated $F$ is defined by $Fu=Nu\_e(0)$, in which $u\_e$ is some extension of $u$ from $C(\Bbb R\_-)$ to $C(\Bbb R)$.
* Furthermore, the fading memory (FM) property is defined for TI operators on $N \in K \subset C(\Bbb R)$ as the existence of some weight function $w: \Bbb R\_+ \to (0,1]$ with $\lim\_{t \to \infty} w(t) = 0$ that makes $K\_-$ closed w.r.t. the weighted norm $w$. i.e. for any $v,u \in K$, one can find a $\delta$ for any $\epsilon$ such that


$$\norm{u-v}\_w:= \sup\_{t\le 0} \abs{u(t)-v(t)}w(-t) < \delta \implies \abs{Nu(0) - Nv(0)} < \epsilon$$


*Lemma 2* states that there are some functionals on $K\_-$ that separate points. To prove this, a class of functionals $G \in \mathbf G \subset K\_-$ is defined by
$$\mathbf{G}=\left\{G(u)=\int\limits\_0^{+\infty}g(\tau)u(-\tau)\dmu\tau :u\in K\_-\land\int\limits\_0^{+\infty}\abs{g(\tau)}w(\tau)^{-1}\dmu\tau\right\}$$
which are (shown to be) continuous w.r.t. the weighted norm $w$ (thanks to the condition introduced above). The authors further construct functions $g\_0$ defined by:


$$g\_0(t):= [u(-t)-v(-t)] w(t)\exp (-t)$$


and their associated $G\_0$ **assuming** that they belong to $\mathbf G$. Then, they show that $g\_0$ indeed separates points on $K\_-$. However, **I don't understand why $G\_0 \in \mathbf G$ in the first place.** $g\_0$, by construction, dependends on $u$ and $v$. Thus, it does not look to me that $G\_0$ is even time-invariant (simply shifting $u$ and $v$ in time will yield a different $g\_0$). So my question is why is the proof of this lemma correct?


**Reference**


[1] Stephen Boyd, Leon O. Chua, "[Fading memory and the problem of approximating nonlinear operators with Volterra series](https://web.stanford.edu/%7Eboyd/papers/pdf/fading_volterra.pdf)" (English), IEEE Transactions on Circuits and Systems 32, 1150-1161 (1985), [MR0809696](https://mathscinet.ams.org/mathscinet-getitem?mr=MR0809696), [Zbl 0587.93028](https://zbmath.org/0587.93028), doi:[10.1109/TCS.1985.1085649](https://doi.org/10.1109/TCS.1985.1085649).



Boyd & Chua 1985: Is the proof of Lemma 2 correct?
Don't let the bad wording of the paper fool you. The statement of lemma 2 in [1] above does not mean that the functional $G$ constructed by the kernel $g\_0$ separates all the points $u, v\in K\_-$ such that $u\neq v$: *it means only that, for any such two functions, you can construct a kernel such that the associated linear functional constructed as above is such that $Gu\neq Gv$*. At first I was fooled by the same thought but then I read *Remark 4* in the same page ([1], §IV p. 1153), which contains the statement quoted below



> 
> Suppose $E$ is a compact metric space and $\mathbf G$ a set of continuous functionals on $E$ which separate points, that is, for any distinct $u,v\in E$ there is a $G\in\mathbf G$ such that $Gu \neq Gv$.
> 
> 
> 


Then I simply realized that the authors want only to show that the functional
$$\DeclareMathOperator{\dmu}{d\!}G\_0(q)=\int\limits\_0^{+\infty}g\_0(\tau)q(-\tau)\dmu\tau \quad \forall q\in K\_-
$$
with the kernel $g\_0$ constructed as described above is such that $G\_0 u\neq G\_0v$ for **any fixed couple of different functions chosen for its construction**. If $u\_1, v\_1\in K\_-$ are a different couple such that $u\_1\neq v\_1$, you do not know if $G\_0u\_1\neq G\_0 v\_1$ (an in general this is not true): however you can construct another functional, say $G\_1$, whose kernel is defined as exactly as $g\_0$ i.e.
$$
g\_1(t):= [u\_1(-t)-v\_1(-t)] w(t)\exp (-t)
$$
and separates the new points but obviously possibly not the points $u$ and $v$.


