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Hilbert's inequality

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In analysis, a branch of mathematics, Hilbert's inequality states that

Contents

| ∑ r ≠ s u r u s ¯ r − s | ≤ π ∑ r | u r | 2 .

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.

Formulation

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

∑ m | u m | 2 < ∞

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

| ∑ r ≠ s u r u s ¯ r − s | ≤ π ∑ r | u r | 2 .

Extensions

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

∑ r ≠ s u r u ¯ s csc ⁡ π ( x r − x s )

and

∑ r ≠ s u r u ¯ s λ r − λ s ,

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

| ∑ r ≠ s u r u s ¯ csc ⁡ π ( x r − x s ) | ≤ δ − 1 ∑ r | u r | 2 .

and

| ∑ r ≠ s u r u s ¯ λ r − λ s | ≤ π τ − 1 ∑ r | u r | 2 .

where

δ = min r , s + ∥ x r − x s ∥ , τ = min r , s + ∥ λ r − λ s ∥ , ∥ s ∥ = min m ∈ Z | s − m |

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

0 < δ r ≤ min s + ∥ x r − x s ∥ and 0 < τ r ≤ min s + ∥ λ r − λ s ∥ ,

then the following inequalities hold:

| ∑ r ≠ s u r u s ¯ csc ⁡ π ( x r − x s ) | ≤ 3 2 ∑ r | u r | 2 δ r − 1 .

and

| ∑ r ≠ s u r u s ¯ λ r − λ s | ≤ 3 2 π ∑ r | u r | 2 τ r − 1 .

References

Hilbert's inequality Wikipedia
(Text) CC BY-SA


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