title: Hilbert's inequality
text: # Hilbert's inequality
  
  In analysis, a branch of mathematics, Hilbert's inequality states that
  
      $\left|\sum _{r\neq s}{\dfrac {u_{r}{\overline {u_{s}}}}{r-s}}\right|\leq \pi \displaystyle \sum _{r}|u_{r}|^{2}.$
  
  for any sequence $u_1,u_2,...$ 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 $ℓ_2.$
  
  ## Formulation
  
  Let $(u_m)$ be a sequence of complex numbers. If the sequence is infinite, assume that it is square-summable:
  
      $\sum _{m}|u_{m}|^{2}<\infty$
  
  Hilbert's inequality (see Steele (2004)) asserts that
  
      $\left|\sum _{r\neq s}{\dfrac {u_{r}{\overline {u_{s}}}}{r-s}}\right|\leq \pi \displaystyle \sum _{r}|u_{r}|^{2}.$
  
  ## Extensions
  
  In 1973, Montgomery & Vaughan reported several generalizations of Hilbert's inequality, considering the bilinear forms
  
      $\sum _{r\neq s}u_{r}{\overline {u}}_{s}\csc \pi (x_{r}-x_{s})$
  
  and
  
      $\sum _{r\neq s}{\dfrac {u_{r}{\overline {u}}_{s}}{\lambda _{r}-\lambda _{s}}},$
  
  where $x_1,x_2,...,x_m$ are distinct real numbers modulo 1 (i.e. they belong to distinct classes in the quotient group R/Z) and $λ_1,...,λ_m$ are distinct real numbers. Montgomery & Vaughan's generalizations of Hilbert's inequality are then given by
  
      $\left|\sum _{r\neq s}u_{r}{\overline {u_{s}}}\csc \pi (x_{r}-x_{s})\right|\leq \delta ^{-1}\sum _{r}|u_{r}|^{2}.$
  
  and
  
      $\left|\sum _{r\neq s}{\dfrac {u_{r}{\overline {u_{s}}}}{\lambda _{r}-\lambda _{s}}}\right|\leq \pi \tau ^{-1}\sum _{r}|u_{r}|^{2}.$
  
  where
  
      $\delta ={\min _{r,s}}_{+}\|x_{r}-x_{s}\|,\quad \tau =\min _{r,s}_{+}\|\lambda _{r}-\lambda _{s}\|,$
      $\|s\|=\min _{m\in \mathbb {Z} }|s-m|$
  
  is the distance from s to the nearest integer, and min+ denotes the smallest positive value. Moreover, if
  
      $0<\delta _{r}\leq {\min _{s}}_{+}\|x_{r}-x_{s}\|\quad {\text{and}}\quad 0<\tau _{r}\leq {\min _{s}}_{+}\|\lambda _{r}-\lambda _{s}\|,$
  
  then the following inequalities hold:
  
      $\left|\sum _{r\neq s}u_{r}{\overline {u_{s}}}\csc \pi (x_{r}-x_{s})\right|\leq {\dfrac {3}{2}}\sum _{r}|u_{r}|^{2}\delta _{r}^{-1}.$
  
  and
  
      $\left|\sum _{r\neq s}{\dfrac {u_{r}{\overline {u_{s}}}}{\lambda _{r}-\lambda _{s}}}\right|\leq {\dfrac {3}{2}}\pi \sum _{r}|u_{r}|^{2}\tau _{r}^{-1}.$
  