Hilbert's inequality


In analysis, a branch of mathematics, Hilbert's inequality states that
for any sequence u1,u2,... of complex numbers. It was first demonstrated by David Hilbert with the constant 2 instead of ; the sharp constant was found by Issai Schur. It implies that the discrete Hilbert transform is a bounded operator in ℓ2.

Formulation

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

Extensions

In 1973, Montgomery & Vaughan reported several generalizations of Hilbert's inequality, considering the bilinear forms
and
where x1,x2,...,xm are distinct real numbers modulo 1 and λ1,...,λm are distinct real numbers. Montgomery & Vaughan's generalizations of Hilbert's inequality are then given by
and
where
is the distance from s to the nearest integer, and min+ denotes the smallest positive value. Moreover, if
then the following inequalities hold:
and