Vol. 15, No. 2, 1965

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ISSN: 0030-8730
Convolution in Fourier-Wiener transform

James Juei-Chin Yeh

Vol. 15 (1965), No. 2, 731–738
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

      ∫
w
G [y] = 0 F [x+ iy]dwx,   y ∈ K.

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

         ∫ 1             ∫ 1
F[x] = ΦF[  α1(t)dx(t),⋅⋅⋅ ,  αn(t)dx(t)]
0               0

where ΦF(ζ1,n) is an entire function of the n complex variables {ζj} of the exponential type and {αj} are n linearly independent real functions of bounded variation on 0 t 1. Let Em 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

            ∫ w   y+x   y−x
(F1 ∗F2)[x] =   F1[21∕2]F2[21∕2]dwy,  x ∈ K.
σ

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

GF1 ∗ GF2[z] = GF1[1z∕2]GF2 [−-z1∕2], z ∈ K.
2        2

Mathematical Subject Classification
Primary: 44.25
Secondary: 42.25
Milestones
Received: 30 March 1964
Published: 1 June 1965
Authors
James Juei-Chin Yeh