Vol. 15, No. 2, 1965

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ISSN: 0030-8730
Convolution transforms whose inversion functions have complex roots

John Dauns and D. V. Widder

Vol. 15 (1965), No. 2, 427–442
Abstract

The convolution transform is defined by the equation

      ∫
∞
f(x) =  −∞G (x− t)φ(t)dt = (G ∗ φ)(x ).
(1.1)

If the kernel G(t) has a bilateral Laplace transform which is the reciprocal of an entire function E(s), then E(s) is called the inversion function of the transform. This terminology is appropriate in view of the fact that the transform (1.1) is inverted, in some sense, by the operator E(D), where D stands for differentiation with respect to x:

E (D )f(x) = φ(x).
(1.2)

It is the purpose of the present paper to prove (1.2) when the roots of E(s) are allowed to be genuinely remote from the real axis.

Mathematical Subject Classification
Primary: 44.25
Milestones
Received: 18 March 1964
Published: 1 June 1965
Authors
John Dauns
D. V. Widder