Asymptotic expansions of Mellin convolutions by means of analytic continuation
The author considers integrals of the form \(I(x) \equiv \int_0^\infty f(t)h(xt) \, dt\). The main aim of this paper is to find asymptotic expansions of \(I(x)\) for large or small \(x\) under mild conditions on \(f\) and \(h\). Many integral transforms (Laplace, Fourier, Stieltjes, Hankel, Poisson, Lambert, etc) can be put in the form \(I(x)\) for adequate functions \(h\), and their asymptotic expansions can be obtained as special cases of the results furnished in this paper. The asymptotic expansion of \(I(x)\) is derived by using analytic continuation, so improving a method initiated by McClure and Wong in 1978. In their work, they obtained explicit error terms for asymptotic expansions of Stieltjes transforms, fractional integrals, Mellin convolutions and other integral transforms. To get such error terms, they used elements of distribution theory. The expansion of \(I(x)\) obtained by López is achieved without the use of distributions, and it is given in terms of two asymptotic sequences, the coefficient of both sequences being Mellin transforms of \(h\) and \(f\). If \(g \in L^1_{loc} (0,\infty )\), the Mellin transform of \(g\) is defined as \(\int_0^z t^{z-1} g(t) \, dt\) (whenever this integral exists), or its analytic continuation as a function of \(z\). The asymptotic character of the expansions in the McCluer-Wong theory is shown when the remainder \(f_n(t)\) of the expansion of \(f(t)\) satisfies the additional hypothesis \(| f_n(t)| \leq c_n t^{-n-b}\) \((0 < t < \infty )\). In the paper under review it is shown that this extra hypothesis is not necessary. Moreover, practical error bounds are provided. The classes of functions \(f, \, h\) (both in \(L^1_{loc} (0,\infty )\)) to which the results are extended are respectively the sets \(F_{b, \alpha}, \, H_{a, \beta }\) (with \(a + \alpha < 1 < b + \beta\), \(\alpha < b\) and \(a < \beta\)) defined as follows: \(f \in F_{b, \alpha}\) (\(h \in H_{a, \beta }\)) if \(f\) (\(h\), resp.) has an asymptotic expansion at \(t = +\infty\) (\(t = 0\), resp.) \(f(t) = \sum_{k=0}^{n-1} A_k t^{-k-b} + f_n(t)\) (\(h(t) = \sum_{k=0}^{n-1} A_k t^{k-a} + h_n(t)\), resp.) for all \(n \geq 1\), where \((A_k)\) is a sequence of complex numbers and \(f_n(t) = O(t^{-n-b})\) as \(t \to \infty\) (\(h_n(t) = O(t^{n-a})\) as \(t \to 0^+\), resp.);\ in addition, \(f(t) = O(t^{-\alpha})\) as \(t \to 0^+\) (\(h(t) = O(t^{-\beta})\) as \(t \to \infty\), resp.).
scientific article; zbMATH DE number 5131529