On the series expansion of certain types of integral transforms—Part I

摘要

The main problem under study is the construction of the complete convergent series development of integral transforms of the kind ∫0+∞f(t)G(xt)dt, x∈R, in which f(t) is known to have a convergent or asymptotic series expansion valid for sufficiently large positive t and G(t) a convergent or asymptotic series expansion valid for sufficiently small positive t, both series mainly consisting of integer or non-integer powers of t. Complementary to this problem is the formulation of associated convergence criterions. In the past, this subject was treated to some extent by Grosjean in the case of Fourier transforms of the sine and the cosine type, and by Vanderleen in the case of Hankel transforms. In the present study, the kernel function G is left arbitrary apart from the specification of the form of its series development in a positive neighbourhood of the origin. In Section 1, the utility of the series expansion of integral transforms is illustrated by means of two examples taken from mathematical physics and references to other examples belonging to different scientific disciplines are given. In Section 2, three expansions theorems and the convergence criterions associated with two of them which were proven resp. by Grosjean and by Vanderleen are recalled in order to facilitate the description of the scope of the set of new articles which will be devoted to the subject. Finally, Section 3 deals with a first generalized series expansion theorem. The generalized moments of f and G appearing in the coefficients are transformed into more tractable integrals under conditions which are usually fulfilled in practice. This leads to a modified version of the first expansion theorem slightly less general than in its original formulation, but better suited for practical application.