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Tauberian theorems for generalized multiplicative convolutions. (English. Russian original) Zbl 0973.46031

Izv. Math. 64, No. 1, 35-92 (2000); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 64, No. 1, 37-94 (2000).
The aim of this paper is best characterized by the authors abstract: “The following problem is discussed. Let \(f\) be a generalized function of slow growth with support on the positive semi-axis, and let \(\varphi_k\) be a sequence of “test” functions such that \(\varphi_k \to\varphi_0\) as \(k\to +\infty\) in some function space. Assume that the following limit exists: \({1\over\rho (k)}(f(kt)\), \(\varphi_k(t)) \to c\), \(k\to +\infty\), where \(\rho(k)\) is a regularly varying function. Find conditions under which the limit \({1\over \rho(k)} (f(kt),\varphi (t))\to c_\varphi\), \(k\to+ \infty\), exists for all test functions \(\varphi\). We state and prove theorems that solve this problem and apply them to the problem of existence of quasi-asymptotics for the solution of an ordinary differential equation with variable coefficients. We prove Abelian and Tauberian theorems for a wide class of integral transformations of distributions, for example, the generalized Stieltjes integral transformation”.

MSC:

46F12 Integral transforms in distribution spaces
40E05 Tauberian theorems
44A15 Special integral transforms (Legendre, Hilbert, etc.)
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