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 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 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