Hilbert's inequality

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

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

for any sequence u₁,u₂,... 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 ℓ₂.

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

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

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

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

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

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

and

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

where x₁,x₂,...,x_m are distinct real numbers modulo 1 (i.e. they belong to distinct classes in the quotient group R/Z) and λ₁,...,λ_m are distinct real numbers. Montgomery & Vaughan's generalizations of Hilbert's inequality are then given by

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

and

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

where

${\displaystyle \delta ={\min _{r,s}}{}_{+}\|x_{r}-x_{s}\|,\quad \tau =\min _{r,s}{}_{+}\|\lambda _{r}-\lambda _{s}\|,}$

${\displaystyle \|s\|=\min _{m\in \mathbb {Z} }|s-m|}$

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

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

then the following inequalities hold:

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

and

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

• Online book chapter Hilbert’s Inequality and Compensating Difficulties extracted from Steele, J. Michael (2004). "Chapter 10: Hilbert's Inequality and Compensating Difficulties". The Cauchy-Schwarz master class: an introduction to the art of mathematical inequalities. Cambridge University Press. pp. 155–165. ISBN 0-521-54677-X..
• Montgomery, H. L.; Vaughan, R. C. (1974). "Hilbert's inequality". J. London Math. Soc. Series 2. 8: 73–82. doi:10.1112/jlms/s2-8.1.73. ISSN 0024-6107.

• Godunova, E.K. (2001) [1994], "Hilbert inequality", Encyclopedia of Mathematics, EMS Press