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 u₁,u₂,... of complex numbers. It was first demonstrated by David Hilbert with the constant 2 $π$ 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}<\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 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

\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}}.

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. .
Montgomery, H. L.; Vaughan, R. C. (1974). "Hilbert's inequality". J. London Math. Soc. (2). 8: 73–82. ISSN 0024-6107.

External links

Godunova, E.K. (2001), "Hilbert inequality", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

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