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
DOI:
https://doi.org/10.1090/S0002-9947-02-03174-4
Published electronically:
October 25, 2002
MathSciNet review:
1938753
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We present a new inversion formula for the classical, finite, and asymptotic Laplace transform of continuous or generalized functions
. The inversion is given as a limit of a sequence of finite linear combinations of exponential functions whose construction requires only the values of
evaluated on a Müntz set of real numbers. The inversion sequence converges in the strongest possible sense. The limit is uniform if
is continuous, it is in
if
, and converges in an appropriate norm or Fréchet topology for generalized functions
. As a corollary we obtain a new constructive inversion procedure for the convolution transform
; i.e., for given
and
we construct a sequence of continuous functions
such that
.
- [Ba] Baeumer, B. A Vector-Valued Operational Calculus and Abstract Cauchy Problems. Dissertation, Louisiana State University, 1997. (http://math.lsu.edu/~tiger )
- [B-L-N] Boris Bäumer, Günter Lumer, and Frank Neubrander, Convolution kernels and generalized functions, Generalized functions, operator theory, and dynamical systems (Brussels, 1997) Chapman & Hall/CRC Res. Notes Math., vol. 399, Chapman & Hall/CRC, Boca Raton, FL, 1999, pp. 68–78. MR 1678671
- [B-N] Boris Bäumer and Frank Neubrander, Laplace transform methods for evolution equations, Confer. Sem. Mat. Univ. Bari 258-260 (1994), 27–60. Swabian-Apulian Meeting on Operator Semigroups and Evolution Equations (Italian) (Ruvo di Puglia, 1994). MR 1385478
- [Do] Doetsch, G. Handbuch der Laplace Transformation. Vol. I-III, Birkhäuser Verlag, Basel-Stuttgart, 1950-1956. MR 13:230f; MR 18:35a; MR 18:894c
- [Fo] C. Foiaş, Approximation des opérateurs de J. Mikusiński par des fonctions continues, Studia Math. 21 (1961/62), 73–74 (French). MR 139907, https://doi.org/10.4064/sm-21-1-73-74
- [L-N] Günter Lumer and Frank Neubrander, Asymptotic Laplace transforms and evolution equations, Evolution equations, Feshbach resonances, singular Hodge theory, Math. Top., vol. 16, Wiley-VCH, Berlin, 1999, pp. 37–57. MR 1691813
- [Mi] Jan Mikusiński, Operational calculus. Vol. 1, 2nd ed., International Series of Monographs in Pure and Applied Mathematics, vol. 109, Pergamon Press, Oxford; PWN—Polish Scientific Publishers, Warsaw, 1983. Translated from the Polish by Janina Smólska; With additional material written in collaboration with Jan Rogut. MR 737380
- [Sk] Kr. Skórnik, On the Foiaş theorem on convolution of continuous functions, Complex analysis and applications ’85 (Varna, 1985) Publ. House Bulgar. Acad. Sci., Sofia, 1986, pp. 604–608. MR 914566
- [Ti] Titchmarsh, E. C. The zeros of certain integral functions. Proceedings of the London Mathematical Society 25 (1926), 283-302.
- [Vi] Vignaux, J. C. Sugli integrali di Laplace asintotici, Atti Accad. naz. Lincei, Rend. Cl. Sci. fis. mat. (6) 29 (1939), 345-400.
Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 44A35, 44A10, 44A40
Retrieve articles in all journals with MSC (2000): 44A35, 44A10, 44A40
Additional Information
Boris Baeumer
Affiliation:
Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin, New Zealand
Email:
bbaeumer@maths.otago.ac.nz
DOI:
https://doi.org/10.1090/S0002-9947-02-03174-4
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