Convolution in Fourier-Wiener transform

James Juei-Chin Yeh

Vol. 15 (1965), No. 2, 731–738

DOI: 10.2140/pjm.1965.15.731

Abstract

Let C be the Wiener space and K be the space of complex valued continuous functions on 0 ≦ t ≦ 1 which vanish at t = 0. The Fourier-Wiener transform of a functional F[x], x ∈ K, is by definition

Let E₀ be the class of functionals F[x] of the type

where Φ_F(ζ₁,⋯,ζ_n) is an entire function of the n complex variables {ζ_j} of the exponential type and {α_j} are n linearly independent real functions of bounded variation on 0 ≦ t ≦ 1. Let E_m be the class of functionals which are mean continuous, entire and of mean exponential type.

We define the convolution of two functionals F₁, F₂ to be

Then if F₁,F₂ ∈ E₀ or F₁,F₂ ∈ E_m, the convolution of F₁, F₂ exists for every x ∈ K and furthermore

Mathematical Subject Classification

Primary: 44.25

Secondary: 42.25

Milestones

Received: 30 March 1964

Published: 1 June 1965