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