Hilbert's inequality

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

[math]\displaystyle{ \left|\sum_{r\neq s}\dfrac{u_{r}\overline{u_{s}}}{r-s}\right|\le\pi\displaystyle\sum_{r}|u_{r}|^2. }[/math]

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:

[math]\displaystyle{ \sum_m |u_m|^2 \lt \infty }[/math]

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

[math]\displaystyle{ \left|\sum_{r\neq s}\dfrac{u_{r}\overline{u_{s}}}{r-s}\right|\le\pi\displaystyle\sum_{r}|u_{r}|^2. }[/math]

Extensions

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

[math]\displaystyle{ \sum_{r\neq s}u_r\overline u_s\csc\pi(x_r-x_s) }[/math]

and

[math]\displaystyle{ \sum_{r\neq s}\dfrac{u_r\overline u_s}{\lambda_r-\lambda_s}, }[/math]

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

[math]\displaystyle{ \left|\sum_{r\neq s} u_r \overline{u_s}\csc\pi(x_r-x_s)\right|\le\delta^{-1}\sum_r |u_r|^2. }[/math]

and

[math]\displaystyle{ \left|\sum_{r\neq s}\dfrac{u_r\overline{u_s}}{\lambda_r-\lambda_s}\right|\le\pi\tau^{-1} \sum_r |u_r|^2. }[/math]

where

[math]\displaystyle{ \delta={\min_{r,s}}{}_{+}\|x_{r}-x_{s}\|, \quad \tau=\min_{r,s}{}_{+}\|\lambda_r-\lambda_s\|, }[/math]

[math]\displaystyle{ \|s\|= \min_{m\in\mathbb{Z}}|s-m| }[/math]

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

[math]\displaystyle{ 0\lt \delta_r \le {\min_s}{}_{+}\|x_r-x_s\| \quad \text{and} \quad 0\lt \tau_{r}\le {\min_{s}}{}_{+}\|\lambda_r-\lambda_s\|, }[/math]

then the following inequalities hold:

[math]\displaystyle{ \left|\sum_{r\neq s} u_r\overline{u_s}\csc\pi(x_r-x_s)\right|\le\dfrac{3}{2} \sum_r |u_r|^2 \delta_r^{-1}. }[/math]

and

[math]\displaystyle{ \left|\sum_{r\neq s}\dfrac{u_r \overline{u_s}}{\lambda_r-\lambda_s}\right|\le \dfrac{3}{2} \pi \sum_r |u_r|^2\tau_r^{-1}. }[/math]

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