Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
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 $\hat f$ 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 $k\star f_{n}\to g$.


References [Enhancements On Off] (What's this?)

  • [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] Bäumer, B., G. Lumer, and F. Neubrander. Convolution kernels and generalized functions. Generalized functions, operator theory, and dynamical systems (Brussels, 1997), 68-78, Res. Notes Math. 399, Chapman & Hall/CRC, Boca Raton, FL, 1999. MR 2000a:44004
  • [B-N] Bäumer, B. and F. Neubrander. Laplace transform methods for evolution equations. Conferenze del Seminario di Matematica dell'Universitá di Bari, 259, 27-60, 1994. MR 97e:47119
  • [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] Foias, C. Approximation des opérateurs de J. Mikusinski par des fonctions continues. Studia Mathematica 21, 73-74, 1961. MR 25:3334
  • [L-N] Lumer, G. and F. Neubrander. Asymptotic Laplace transforms and evolution equations. Evolution equations, Feshbach resonances, singular Hodge theory, 37-57, Math. Top. 16, Wiley-VCH, Berlin, 1999. MR 2000f:47068
  • [Mi] Mikusinski, J. Operational Calculus. v. 1-2, Pergamon Press, 2nd edition, 1987. MR 86b:44017; MR 88k:44010
  • [Sk] Skórnik, K. On the Foias theorem on convolution of continuous functions. Complex Analysis and Applications '85 (Varna, 1985), 604-608, Publ. House Bulgar. Acad. Sci., Sofia, 1986. MR 89j:46057
  • [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.

Similar Articles

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: http://dx.doi.org/10.1090/S0002-9947-02-03174-4
PII: S 0002-9947(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