In analysis, a branch of mathematics, Hilbert's inequality states that
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≤
π
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u
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2
.
{\displaystyle \left|\sum _{r\neq s}{\dfrac {u_{r}{\overline {u_{s}}}}{r-s}}\right|\leq \pi \displaystyle \sum _{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.
Contents
Formulation
Let (um) be a sequence of complex numbers. If the sequence is infinite, assume that it is square-summable:
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u
m
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2
<
∞
{\displaystyle \sum _{m}|u_{m}|^{2}<\infty }
Hilbert's inequality (see Steele (2004)) asserts that
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u
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u
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¯
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≤
π
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u
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Extensions
In 1973, Montgomery & Vaughan reported several generalizations of Hilbert's inequality, considering the bilinear forms
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csc
π
(
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s
)
{\displaystyle \sum _{r\neq s}u_{r}{\overline {u}}_{s}\csc \pi (x_{r}-x_{s})}
and
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u
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λ
r
−