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Transactions of the American Mathematical Society

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On the inversion of the convolution and Laplace transform

Author: Boris Baeumer
Journal: Trans. Amer. Math. Soc. 355 (2003), 1201-1212
MSC (2000): Primary 44A35, 44A10, 44A40
Published electronically: October 25, 2002
MathSciNet review: 1938753
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Abstract: We present a new inversion formula for the classical, finite, and asymptotic Laplace transform $\hat f$ of continuous or generalized functions $f$. The inversion is given as a limit of a sequence of finite linear combinations of exponential functions whose construction requires only the values of $\hat f$ evaluated on a Müntz set of real numbers. The inversion sequence converges in the strongest possible sense. The limit is uniform if $f$ is continuous, it is in $L^{1}$ if $f\in L^{1}$, and converges in an appropriate norm or Fréchet topology for generalized functions $f$. As a corollary we obtain a new constructive inversion procedure for the convolution transform ${\mathcal K}:f\mapsto k\star f$; i.e., for given $g$ and $k$ we construct a sequence of continuous functions $f_{n}$ such that $k\star f_{n}\to g$.

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Additional Information

Boris Baeumer
Affiliation: Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin, New Zealand
MR Author ID: 688464

Keywords: Operational calculus, generalized functions, integral transforms.
Received by editor(s): January 25, 1999
Received by editor(s) in revised form: August 5, 2002
Published electronically: October 25, 2002
Article copyright: © Copyright 2002 American Mathematical Society