Some Remarks on Analytic Representations and Products of Distributions

Abstract

H. J. BREMERMANNt 1. Representation of distributions by analytic functions of one complex variable. This section is a brief summary; most of the proofs may be found in Bremermann [5]. Compare also Beltrami-Wohlers [1]. Terminology and notations are consistent with L. Schwartz [12]. Let T be a distribution in (&'(E1)), where E1 denotes the reals, or T C (O'L2); then the function T'(z) = (1/2ri)(T, 1/t -z) is an analytic function for Im z # 0 (Im z imaginary part of z). T(z) represents T in the following sense: T(x + iE) T(x iE) converges to T for E -> 0+ in the topology of (O'). We call T(z) the Cauchy representation of T. For T C (D'(El)) the function (t z)f1 is not a test function, T(z), in general, does not exist, but the following result holds: there exists a pair of functions f+(z) and f_(z), analytic in the upper and lower half-plane, respectively, such thatf+(x + iE) f(x ie) converges to T for e -> 0+ in the topology of (OD'). We call f+ the forward function, f_ the backward function and the pair an analytic representation of T. Analytic representations of the same distribution differ by at most an eintire function [14], [24]. If a pair of analytic functions represents T, then their complex derivatives represent T'. If the complement of the support of a distribution is nonempty, then for any analytic representation of T the forward and backward functions f+ and f_ are analytic continuations of each other and remain analytic on the complement of the support of T. Not every pair of functions, analytic in the upper and lower half-planes, respectively, represents a distribution: f+(z) = f_(z) = exp (1/z2) is a counter-example. A tempered function is a continuous complex-valued function oni El that grows at infinity no faster than a polynomial. For a tempered function f