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, suppose that u is an L1 function with Fourier series


 * $$u(t) = \tfrac12 a_0 + \sum_{n=1}^\infty \left(a_n\cos n t + b_n\sin n t\right),$$

then the conjugate of u is defined by the series


 * $$v(t) = \sum_{n=1}^\infty \left(a_n\sin n t - b_n\cos n t\right).$$