Hilbert's inequality
# Hilbert's inequality

In analysis, a branch of mathematics, Hilbert's inequality states that

    $\left|\sum _{r\neq s}{\dfrac {u_{r}{\overline {u_{s}}}}{r-s}}\right|\leq \pi \displaystyle \sum _{r}|u_{r}|^{2}.$

for any sequence $u_1,u_2,...$ of complex numbers. It was first demonstrated by David Hilbert with the constant 2π instead of π; the sharp constant was found by Issai Schur. It implies that the discrete Hilbert transform is a bounded operator in $ℓ_2.$

## Formulation

Let $(u_m)$ be a sequence of complex numbers. If the sequence is infinite, assume that it is square-summable:

    $\sum _{m}|u_{m}|^{2}<\infty$

Hilbert's inequality (see Steele (2004)) asserts that

    $\left|\sum _{r\neq s}{\dfrac {u_{r}{\overline {u_{s}}}}{r-s}}\right|\leq \pi \displaystyle \sum _{r}|u_{r}|^{2}.$

## Extensions

In 1973, Montgomery & Vaughan reported several generalizations of Hilbert's inequality, considering the bilinear forms

    $\sum _{r\neq s}u_{r}{\overline {u}}_{s}\csc \pi (x_{r}-x_{s})$

and

    $\sum _{r\neq s}{\dfrac {u_{r}{\overline {u}}_{s}}{\lambda _{r}-\lambda _{s}}},$

where $x_1,x_2,...,x_m$ are distinct real numbers modulo 1 (i.e. they belong to distinct classes in the quotient group R/Z) and $λ_1,...,λ_m$ are distinct real numbers. Montgomery & Vaughan's generalizations of Hilbert's inequality are then given by

    $\left|\sum _{r\neq s}u_{r}{\overline {u_{s}}}\csc \pi (x_{r}-x_{s})\right|\leq \delta ^{-1}\sum _{r}|u_{r}|^{2}.$

and

    $\left|\sum _{r\neq s}{\dfrac {u_{r}{\overline {u_{s}}}}{\lambda _{r}-\lambda _{s}}}\right|\leq \pi \tau ^{-1}\sum _{r}|u_{r}|^{2}.$

where

    $\delta ={\min _{r,s}}_{+}\|x_{r}-x_{s}\|,\quad \tau =\min _{r,s}_{+}\|\lambda _{r}-\lambda _{s}\|,$
    $\|s\|=\min _{m\in \mathbb {Z} }|s-m|$

is the distance from s to the nearest integer, and min+ denotes the smallest positive value. Moreover, if

    $0<\delta _{r}\leq {\min _{s}}_{+}\|x_{r}-x_{s}\|\quad {\text{and}}\quad 0<\tau _{r}\leq {\min _{s}}_{+}\|\lambda _{r}-\lambda _{s}\|,$

then the following inequalities hold:

    $\left|\sum _{r\neq s}u_{r}{\overline {u_{s}}}\csc \pi (x_{r}-x_{s})\right|\leq {\dfrac {3}{2}}\sum _{r}|u_{r}|^{2}\delta _{r}^{-1}.$

and

    $\left|\sum _{r\neq s}{\dfrac {u_{r}{\overline {u_{s}}}}{\lambda _{r}-\lambda _{s}}}\right|\leq {\dfrac {3}{2}}\pi \sum _{r}|u_{r}|^{2}\tau _{r}^{-1}.$
