Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Asymptotic inversion of Laplace transforms: a class of counterexamples

Author: John S. Lew
Journal: Proc. Amer. Math. Soc. 39 (1973), 329-336
MSC: Primary 44A10
MathSciNet review: 0324325
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ f$ be a complex-valued locally integrable function on $ [0, + \infty )$, and let $ Lf$ be its Laplace transform, whenever and wherever it exists. We review some known methods, exact and approximate, for recovering $ f$ from $ Lf$. Since numerical algorithms need auxiliary information about $ f$ near $ + \infty $, we note that the behavior of $ f$ near $ + \infty $ depends on the behavior of $ Lf$ near 0 +, hence that our ability to retrieve $ f$ is limited by the class of momentless functions, namely, all functions $ f$ such that $ Lf(s)$ converges absolutely for $ \operatorname{Re} (s) > 0$ and satisfies

$\displaystyle Lf(s) = o({s^n}){\text{ near }}0 + \quad {\text{for}}\;n = 0,1,2, \cdots .$

We discuss the space $ Z$ of momentless functions and complex distributions, then construct a family of elements in this space which defy various plausible conjectures.

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

  • [1] Richard Bellman, Robert E. Kalaba, and Jo Ann Lockett, Numerical inversion of the Laplace transform: Applications to biology, economics, engineering and physics, American Elsevier Publishing Co., Inc., New York, 1966. MR 0205454
  • [2] J. W. Cooley, P. A. W. Lewis and P. D. Welch, The fast Fourier transform algorithm: programming considerations in the calculation of sine, cosine, and Laplace transforms, J. Sound Vib. 12 (1970), 315-337.
  • [3] Gustav Doetsch, Handbuch der Laplace-Transformation. Band I. Theorie der Laplace-Transformation, Verlag Birkhäuser, Basel, 1950 (German). MR 0043253
  • [4] Gustav Doetsch, Handbuch der Laplace-Transformation, Birkhäuser Verlag, Basel-Stuttgart, 1972 (German). Band II: Anwendungen der Laplace-Transformation. 1. Abteilung; Verbesserter Nachdruck der ersten Auflage 1955; Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften. Mathematische Reihe, Band 15. MR 0344808
  • [5] A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of integral transforms. Vol. I, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954. Based, in part, on notes left by Harry Bateman. MR 0061695
  • [6] H. Goldenberg, The evaluation of inverse Laplace transforms without the aid of contour integration, SIAM Rev. 4 (1962), 94–104. MR 0136937,
  • [7] Richard A. Handelsman and John S. Lew, Asymptotic expansion of Laplace transforms near the origin, SIAM J. Math. Anal. 1 (1970), 118–130. MR 0259504,
  • [8] -, Asymptotic expansion of Laplace convolutions for large argument, SIAM Rev. 13 (1971), 269.
  • [9] Richard A. Handelsman and John S. Lew, Asymptotic expansion of Laplace convolutions for large argument and tail densities for certain sums of random variables, SIAM J. Math. Anal. 5 (1974), 425–451. MR 0344766,
  • [10] Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957. rev. ed. MR 0089373
  • [11] T. E. Hull and C. Froese, Asymptotic behaviour of the inverse of a Laplace transform, Canad. J. Math. 7 (1955), 116–125. MR 0066485,
  • [12] Jean Lavoine, Sur des théorèmes abéliens et taubériens de la transformation de Laplace, Ann. Inst. H. Poincaré Sect. A (N.S.) 4 (1966), 49–65 (French). MR 0206634
  • [13] H. Mellin, Abriss einer allgemeinen und einheitlichen Theorie der asymptotische Reihen, Wissenschaftliche Vorträge gehalten auf dem 5 Kongress der Skandinav. Mathematiker in Helsingfors, 4-7 Juli 1922, vol. 1, 1922, pp. 1-17.
  • [14] Walter Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley and Sons), New York-London, 1962. MR 0152834
  • [15] Laurent Schwartz, Mathematics for the physical sciences, Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966. MR 0207494
  • [16] G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R.I., 1959. MR 21 #5029.
  • [17] Eberhard Wagner, Taubersche Sätze reeller Art für die Laplace-transformation, Math. Nachr. 31 (1966), 153–168 (German). MR 0194797,
  • [18] Eberhard Wagner, Ein reeller Tauberscher Satz für die Laplace-Transformation, Math. Nachr. 36 (1968), 323–331 (German). MR 0233113,
  • [19] D. V. Widder, The Laplace transform, Princeton Math. Series, vol. 6, Princeton Univ. Press, Princeton, N.J., 1941. MR 3, 232.
  • [20] D. V. Widder, Inversion of a heat transform by use of series, J. Analyse Math. 18 (1967), 389–413. MR 0211206,
  • [21] -, An introduction to transform theory, Pure and Appl. Math., vol. 42, Academic Press, New York, 1971.
  • [22] A. H. Zemanian, Distribution theory and transform analysis. An introduction to generalized functions, with applications, McGraw-Hill Book Co., New York-Toronto-London-Sydney, 1965. MR 0177293
  • [23] V. Riekstynja, Generalized asymptotic expansions for a contour integral, Latvijas Valsts Univ. Zinātn. Raksti 28 (1959), 111–126 (Russian, with Latvian summary). MR 0123154
  • [24] -, Asymptotic expansions of some integrals and the sums of power series, Latvian Math. Yearbook 9 (1971), 203-220.
  • [25] Eberhard Wagner, Taubersche Sätze reeler Art für Integraltransformationen mit Kernen der Form expℎ(𝑠)𝑡, Wiss. Z. Univ. Rostock Math.-Naturwiss. Reihe 20 (1971), no. 5/6, 313–320 (German, with Russian, English and French summaries). MR 0425522

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 44A10

Retrieve articles in all journals with MSC: 44A10

Additional Information

Keywords: Laplace transform, asymptotic inversion, inverse Abelian theorem, Tauberian theorem, Mellin series
Article copyright: © Copyright 1973 American Mathematical Society