Convolution transforms whose inversion functions have complex roots
John Dauns, David Vernon Widder · Pacific Journal of Mathematics · 1965
The convolution transform is defined by the equation (1.1) f{x) = f" G(x -t)φ(t)dt = (G*φ)(x) .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: (1.2) E(D)f(x) = φ{x) .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.Formula (1.2) was first proved by Widder [7] in 1947 for a large class of entire functions E(s) and by Hirschman and Widder [3] in 1949 for the whole Laguerre-Pόlya class.The latter functions have real roots only, indeed are the uniform limits of polynomials with real roots only, see p. 42 of [5].In 1951 Hirschman and Widder [4] extended this inversion theory, allowing the roots of E(s) to be complex.However, the roots were asymptotically real in the sense that their arguments clustered to 0 or to 7Γ.At the same time A. 0. Garder [2] allowed the approach to the real axis to be slower.We require only that they should occur in pairs symmetric in the origin and in a sector inside the sector tan (arg s) \ < 1.More precisely:We wish also to call attention to some new asymptotic relations.