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

 

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:

Hilbert's inequality (see ) asserts that

Extensions

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

 

and

 

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

 

and

 

where

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

then the following inequalities hold:

 

and

References

 Online book chapter Hilbert’s Inequality and Compensating Difficulties extracted from . External links
 

Inequalities
Complex analysis
Number theory