Hilbert's inequality

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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 2π instead of π; the sharp constant was found by Issai Schur. It implies that the discrete Hilbert transform is a bounded operator in 2.


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

Hilbert's inequality (see Steele (2004)) asserts that


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


where x1,x2,...,xm are distinct real numbers modulo 1 (i.e. they belong to distinct classes in the quotient group R/Z) and λ1,...,λm are distinct real numbers. Montgomery & Vaughan's generalizations of Hilbert's inequality are then given by



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

then the following inequalities hold:



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