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 ℓ2.
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:
- Online book chapter Hilbert’s Inequality and Compensating Difficulties extracted from Steele, J. Michael (2004). "Chapter 10: Hilbert's Inequality and Compensating Difficulties". The Cauchy-Schwarz master class: an introduction to the art of mathematical inequalities. Cambridge University Press. pp. 155–165. ISBN 0-521-54677-X..
- Montgomery, H. L.; Vaughan, R. C. (1974). "Hilbert's inequality". J. London Math. Soc. Series 2. 8: 73–82. ISSN 0024-6107.