The Parseval formulae for monotonic functions. I

Abstract

The Parseval formulae for Fourier cosine and sine transforms,are of course most widely known in connexion with the classical theorems of Plancherel on functions of the class L2 (whose transforms are defined by mean convergence), and with their generalizations. We cannot expect to obtain anything as elegant as the ‘L2’ results when we consider (1) for functions of other kinds. Nevertheless, since the most obvious way of defining Fourier transforms is by means of Lebesgue or Cauchy integrals, we naturally wish to know how far the formulae (1) hold good for transforms obtained in this way. The two most familiar classes of functions having such transforms are:(i) functions f(t) integrable in the Lebesgue sense in (0, ∞), whose transforms Fe(x) and Fs(x) are defined by the Lebesgue integrals respectively; and(ii) functions f(t) which decrease in (0, ∞), tend to zero as t → ∞, and are integrable over any finite interval (0, T); in this case the transforms are defined by the Cauchy integrals .