## 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.)
Full Text: