The hilbert transform of periodic distributions

Abstract

Let f(t) be a periodic function with period 2τ defined on the real line R. A new definition of the Hilbert transform (Hf)(x) of this function is given by provided this limit exists; the integral in (1) is taken in the Cauchy principal-value sense. It is shown that the definition (1) is equivalent to the conventional definition, where the integral in (2) is taken in the Cauchy principal value sense too. We then use our results to extend the Hilbert transform to periodic distributions of period 2τ defined on the real line, i.e. to is the space of infinitely differentiable functions with period 2τ defined on the real line. An inversion formula over the space is proved. Amongst many other related results which are proved, we have proved the existence of a periodic harmonic function u(x y) (periodic in x with period 2τ) whose boundary values at a semicircular boundary are given.