Hilbert's inequality

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

$\left|\sum _{r\ne s}\frac{u_r\overline{u_s}}{r-s}\right|\le \pi \sum _r|u_r{|}^2.$

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.

Formulation

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

$\sum _m|u_m{|}^2<\mathrm{\infty }$

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

$\left|\sum _{r\ne s}\frac{u_r\overline{u_s}}{r-s}\right|\le \pi \sum _r|u_r{|}^2.$

Extensions

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

$\sum _{r\ne s}{u}_r{\bar{u}}_s\csc \pi (x_r-x_s)$

and

$\sum _{r\ne s}\frac{u_r{\bar{u}}_s}{{\lambda }_r-{\lambda }_s},$

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

$|\sum _{r\ne s}{u}_r\overline{u_s}\csc \pi (x_r-x_s)|\le {\delta }^{-1}\sum _r|u_r{|}^2.$

and

$\left|\sum _{r\ne s}\frac{u_r\overline{u_s}}{{\lambda }_r-{\lambda }_s}\right|\le \pi {\tau }^{-1}\sum _r|u_r{|}^2.$

where

$\delta =\underset{r,s}{\min }{}_{+}\Vert x_r-x_s\Vert ,\tau =\underset{r,s}{\min }{}_{+}\Vert {\lambda }_r-{\lambda }_s\Vert ,$

$\Vert s\Vert =\underset{m\in \mathbb{\mathbb{Z}}}{\min }|s-m|$

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

$0<{\delta }_r\le \underset{s}{\min }{}_{+}\Vert x_r-x_s\Vert \text{and}0<{\tau }_r\le \underset{s}{\min }{}_{+}\Vert {\lambda }_r-{\lambda }_s\Vert ,$

then the following inequalities hold:

$|\sum _{r\ne s}{u}_r\overline{u_s}\csc \pi (x_r-x_s)|\le \frac{3}{2}\sum _r|u_r{|}^2{\delta }_r^{-1}.$

and

$\left|\sum _{r\ne s}\frac{u_r\overline{u_s}}{{\lambda }_r-{\lambda }_s}\right|\le \frac{3}{2}\pi \sum _r|u_r{|}^2{\tau }_r^{-1}.$

References

• 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. https://books.google.com/books?id=bvgBdZKEYAEC&pg=PA155. .
• 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. 

External links

• Hazewinkel, Michiel, ed. (2001), "Hilbert inequality", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Hilbert_inequality