TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIElY Volume 272, Number 2, August 1982 ASYMPTOTIC EXPANSIONS OF SOME INTEGRAL TRANSFORMS BY USING GENERALIZED FUNCTIONS BY AHMED I, ZAYED ABSTRACT. The technique devised by Wong to derive the asymptotic expansions of multiple Fourier transforms by using the theory of Schwartz distributions is ex- tended 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. 1. Introduction. Recently, the theory of generalized functions has been introduced into the field of asymptotic expansions of integral transforms; see [18, 19,6]. One of the advantages of this approach is to be able to interpret and to assign values to some divergent integrals. This approach goes back to Lighthill [10] who was the first to use generalized functions to study the asymptotic expansions of Fourier transforms. Borrowing Lighthill's idea, several people were able to derive asymptotic expan- sions for other types of integral transforms such as Laplace [2], Stieltjes [11], Riemann-Liouville fractional integral transforms [12] and Mellin convolution of two algebraically dominated functions [17]. The use of generalized functions in the theory of asymptotic expansions has another advantage; that is, in some cases it can provide explicit expressions for the error terms in shorter and more direct ways than the classical methods [9,11]. However, one should not overemphasize the importance of the generalized functions approach since, for the most part, it is an alternative but not an exclusive technique. The use of generalized functions, except for the case of Fourier transform, was basically limited to integral transforms with exponentially decaying kernels. In a recent paper [18], Wong devised a new technique based on the theory of Schwartz distributions to provide an alternative derivation to the one given in [15] for the asymptotic expansions of multiple Fourier transforms. In doing that, he actually showed that the generalized functions method could be applied to integral transforms with oscillatory kernels with equal advantage to the classical methods. The analysis of his technique shows that the theory of Schwartz distributions and their Fourier transforms [14] plays a vital role in the proof. Therefore, it seems inevitable that extending Wong's technique to other types of integral transforms Received by the editors March 9, 1981 and, in revised form, July 20, 1981. 1980 Mathematics Subject Classification. Primary 41A60; Secondary 46F12. Key words and phrases. Asymptotic expansion, generalized function, integral transform. 785 © 1982 American Mathematical Society 0002-9947/81/0000-0657/$06.00 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use