Asymptotic expansions of some integral transforms by using generalized functions
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 by Ahmed I. Zayed PDF
 Trans. Amer. Math. Soc. 272 (1982), 785802 Request permission
Abstract:
The technique devised by Wong to derive the asymptotic expansions of multiple Fourier transforms by using the theory of Schwartz distributions is extended to a large class of integral transforms. The extension requires establishing a general procedure to extend these integral transforms to generalized functions. Wong’s technique is then applied to some of these integral transforms to obtain their asymptotic expansions. This class of integral transforms encompasses, among others, the Laplace, the Airy, the $K$ and the Hankel transforms.References

N. Bleistein and R. A. Handelsman, Asymptotic expansions of integrals, Holt, Rinehart and Winston, New York, 1975.
 P. Durbin, Asymptotic expansion of Laplace transforms about the origin using generalized functions, J. Inst. Math. Appl. 23 (1979), no. 2, 181–192. MR 529364
 I. M. Gel’fand and G. E. Shilov, Generalized functions. Vol. I: Properties and operations, Academic Press, New YorkLondon, 1964. Translated by Eugene Saletan. MR 0166596
 Richard A. Handelsman and John S. Lew, Asymptotic expansion of a class of integral transforms with algebraically dominated kernels, J. Math. Anal. Appl. 35 (1971), 405–433. MR 278012, DOI 10.1016/0022247X(71)902277
 Witold Hurewicz, Lectures on ordinary differential equations, Technology Press of The Massachusetts Institute of Technology, Cambridge, Mass.; John Wiley & Sons, Inc., New York, 1958. MR 0090703
 D. S. Jones, Generalised functions, McGrawHill Book Co., New YorkToronto, Ont.London, 1966. MR 0217534 E. L. Koh and A. H. Zemanian, The complex Hankel and $I$transformations of generalized functions, SIAM J. Appl. Math. 16 (1968), 945957. G. Köthe, Topological vector spaces, vol. 1, SpringerVerlag, New York, 1969.
 H. A. Lauwerier, Asymptotic analysis, Mathematical Centre Tracts, No. 54, Mathematisch Centrum, Amsterdam, 1974. MR 0467123 M. J. Lighthill, Fourier analysis and generalized functions, Cambridge Univ. Press, Cambridge, 1958.
 J. P. McClure and R. Wong, Explicit error terms for asymptotic expansions of Stieltjes transforms, J. Inst. Math. Appl. 22 (1978), no. 2, 129–145. MR 509152
 J. P. McClure and R. Wong, Exact remainders for asymptotic expansions of fractional integrals, J. Inst. Math. Appl. 24 (1979), no. 2, 139–147. MR 544430
 F. W. J. Olver, Asymptotics and special functions, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New YorkLondon, 1974. MR 0435697
 Laurent Schwartz, Théorie des distributions, Publications de l’Institut de Mathématique de l’Université de Strasbourg, IXX, Hermann, Paris, 1966 (French). Nouvelle édition, entiérement corrigée, refondue et augmentée. MR 0209834
 P. N. Shivakumar and R. Wong, Asymptotic expansion of multiple Fourier transforms, SIAM J. Math. Anal. 10 (1979), no. 6, 1095–1104. MR 547798, DOI 10.1137/0510100
 R. Wong, Error bounds for asymptotic expansions of Hankel transforms, SIAM J. Math. Anal. 7 (1976), no. 6, 799–808. MR 415224, DOI 10.1137/0507061
 R. Wong, Explicit error terms for asymptotic expansions of Mellin convolutions, J. Math. Anal. Appl. 72 (1979), no. 2, 740–756. MR 559402, DOI 10.1016/0022247X(79)902610
 R. Wong, Distributional derivation of an asymptotic expansion, Proc. Amer. Math. Soc. 80 (1980), no. 2, 266–270. MR 577756, DOI 10.1090/S00029939198005777568
 R. Wong, Error bounds for asymptotic expansions of integrals, SIAM Rev. 22 (1980), no. 4, 401–435. MR 593856, DOI 10.1137/1022086
 A. H. Zemanian, The distributional Laplace and Mellin transformations, SIAM J. Appl. Math. 14 (1966), 41–59. MR 190640, DOI 10.1137/0114004 —, Generalized integral transformations, Interscience, New York, 1966.
 D. S. Jones, Generalized transforms and their asymptotic behaviour, Philos. Trans. Roy. Soc. London Ser. A 265 (1969), 1–43. MR 249950, DOI 10.1098/rsta.1969.0039
Additional Information
 © Copyright 1982 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 272 (1982), 785802
 MSC: Primary 41A60; Secondary 44A05, 46F12
 DOI: https://doi.org/10.1090/S00029947198206620679
 MathSciNet review: 662067