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mathematical analysis > complex analysis > Hilbert's inequality

Preferred term

Hilbert's inequality

Definition(s)

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

${\\displaystyle \\left|\\sum _{r\ eq s}{\\dfrac {u_{r}{\\overline {u_{s}}}}{r-s}}\ ight|\\leq \\pi \\displaystyle \\sum _{r}|u_{r}|^{2}.}$

for any sequence $u 1, u 2,...$ of complex numbers. It was first demonstrated by David Hilbert with the constant $2 π$ instead of $π$ ; the sharp constant was found by Issai Schur. It implies that the discrete Hilbert transform is a bounded operator in $ℓ 2$ .

(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Hilbert%27s_inequality)

Broader concept(s)

complex analysis

In other languages

inégalité de Hilbert

French

URI

http://data.loterre.fr/ark:/67375/PSR-NL0VXQGH-N

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