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Hilbert's inequality

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

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

Formulation

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

∑ m | u m | 2 < ∞ {\displaystyle \sum _{m}|u_{m}|^{2}<\infty }

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

| ∑ r ≠ s u r u s ¯ r − s | ≤ π ∑ r | u r | 2 . {\displaystyle \left|\sum _{r\neq s}{\dfrac {u_{r}{\overline {u_{s}}}}{r-s}}\right|\leq \pi \displaystyle \sum _{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 ) {\displaystyle \sum _{r\neq s}u_{r}{\overline {u}}_{s}\csc \pi (x_{r}-x_{s})}

and

∑ r ≠ s u r u ¯ s λ r − λ s , {\displaystyle \sum _{r\neq s}{\dfrac {u_{r}{\overline {u}}_{s}}{\lambda _{r}-\lambda _{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 . {\displaystyle \left|\sum _{r\neq s}u_{r}{\overline {u_{s}}}\csc \pi (x_{r}-x_{s})\right|\leq \delta ^{-1}\sum _{r}|u_{r}|^{2}.}

and

| ∑ r ≠ s u r u s ¯ λ r − λ s | ≤ π τ − 1 ∑ r | u r | 2 . {\displaystyle \left|\sum _{r\neq s}{\dfrac {u_{r}{\overline {u_{s}}}}{\lambda _{r}-\lambda _{s}}}\right|\leq \pi \tau ^{-1}\sum _{r}|u_{r}|^{2}.}

where

δ = min r , s + ‖ x r − x s ‖ , τ = min r , s + ‖ λ r − λ s ‖ , {\displaystyle \delta ={\min _{r,s}}{}_{+}\|x_{r}-x_{s}\|,\quad \tau =\min _{r,s}{}_{+}\|\lambda _{r}-\lambda _{s}\|,}
‖ s ‖ = min m ∈ Z | s − m | {\displaystyle \|s\|=\min _{m\in \mathbb {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 ‖ , {\displaystyle 0<\delta _{r}\leq {\min _{s}}{}_{+}\|x_{r}-x_{s}\|\quad {\text{and}}\quad 0<\tau _{r}\leq {\min _{s}}{}_{+}\|\lambda _{r}-\lambda _{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 . {\displaystyle \left|\sum _{r\neq s}u_{r}{\overline {u_{s}}}\csc \pi (x_{r}-x_{s})\right|\leq {\dfrac {3}{2}}\sum _{r}|u_{r}|^{2}\delta _{r}^{-1}.}

and

| ∑ r ≠ s u r u s ¯ λ r − λ s | ≤ 3 2 π ∑ r | u r | 2 τ r − 1 . {\displaystyle \left|\sum _{r\neq s}{\dfrac {u_{r}{\overline {u_{s}}}}{\lambda _{r}-\lambda _{s}}}\right|\leq {\dfrac {3}{2}}\pi \sum _{r}|u_{r}|^{2}\tau _{r}^{-1}.}

References

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