User:Silly rabbit/Hilbert

The Hilbert transform provides a convenient means to determine the Fourier conjugate to a function u. Working over the circle, following Zygmund (1968), suppose that u is an L¹ function with Fourier series

$u(t)={\tfrac {1}{2}}a_{0}+\sum _{n=1}^{\infty }\left(a_{n}\cos nt+b_{n}\sin nt\right),$

then the conjugate of u is defined by the series

$v(t)=\sum _{n=1}^{\infty }\left(a_{n}\sin nt-b_{n}\cos nt\right).$