Hilbert's inequality explained

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

\left|\sum _{r ≠} \dfrac{u _{r} \overline{u _{s} }}\right|\le\pi\displaystyle\sum_|u_|^2.

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

Formulation

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

\sum _{m} |u ^{2} _{m|} < inf ty

Hilbert's inequality (see) asserts that

\left|\sum _{r ≠} \dfrac{u _{r} \overline{u _{s} }}\right|\le\pi\displaystyle\sum_|u_|^2.

Extensions

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

\sum _{r ≠} u _{r\overline} u _{s\csc\pi(x} _{r-x} _{s)}

and

\sum _{r ≠} \dfrac{u _{r\overline} u _{s}{λ} _{r-λ} _{s},}

where are distinct real numbers modulo 1 (i.e. they belong to distinct classes in the quotient group) and are distinct real numbers. Montgomery & Vaughan's generalizations of Hilbert's inequality are then given by

\left|\sum _{r ≠} u _{r} \overline{u _{s}\csc\pi(x} _{r-x} ^{-1} ^{s)\right|\le\delta} \sum _{r} |u ^{2 .} _{r|}

and

\left|\sum _{r ≠} \dfrac{u _{r\overline{u} _{s}}{λ} _{r-λ} ^{-1} ^{s}\right|\le\pi\tau} \sum _{r} |u ^{2 .} _{r|}

where

\delta = { min _{r,s} }_\|x_-x_\|, \quad \tau=\min__\|\lambda_r-\lambda_s\|, \|s\| = min _{m\inZ} |s - m|

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

0 <\delta _{r} \le { min _{s}{}} _{+} \|x _{r-x} _{s\|} and 0 <\tau _{r} \le { min _{s} }_\|\lambda_r-\lambda_s\|,

then the following inequalities hold:

\left|\sum _{r ≠} u _{r\overline{u} _{s}\csc\pi(x} _{r-x} _{s)\right|\le\dfrac{3}{2}} \sum _{r} |u ^{2} _{r|} \delta ^{-1} _{r} .

and

\left|\sum _{r ≠} \dfrac{u _{r} \overline{u _{s}}{λ} _{r-λ} _{s}\right|\le} \dfrac{ 3 }{ 2 } \pi \sum _{r} |u ^{-1} _{r} .

References

• Online book chapter Hilbert’s Inequality and Compensating Difficulties extracted from Book: Steele, J. Michael. The Cauchy-Schwarz master class: an introduction to the art of mathematical inequalities . 155–165 . Chapter 10: Hilbert’s Inequality and Compensating Difficulties . https://books.google.com/books?id=bvgBdZKEYAEC&pg=PA155 . 2004 . 0-521-54677-X . Cambridge University Press. .
• Montgomery. H. L.. Hugh Montgomery (mathematician). Vaughan. R. C.. Hilbert's inequality. J. London Math. Soc. . Series 2. 8. 1974. 73–82. 10.1112/jlms/s2-8.1.73. 0024-6107. MV.