Multidimensional Tauberian theorems for vector-valued distributions (Q2815241)

The paper is devoted to the study of the rich subject of Tauberian theory of generalized functions in the multidimensional setting. The authors reconsider important contributions of Vladimirov, Drozhzhinov and Zav'yalov (see the list of references) and extend their approach to the biggest class of kernels determined by non-degenerate conditions. The results are formulated for a general class of Banach space-valued distributions. The main asymptotic properties are given in terms of the so-called quasiasymptotic behavior and quasiasymptotic boundedness of certain degree.NEWLINENEWLINE The motivation for the study, together with a list of results, is exposed in a well-written introduction. Section 2 consists of definitions of spaces and quasiasymptotics, while Section 3 contains abelian-type results as a preparation for the main Tauberian theorems exposed in Section 4 and proved in Section 6. Section 4 also contains informative remarks which explain the connection between the main results and related results in the literature. In Section 5, the authors apply their technique to the wavelet analysis and link their non-degenerate condition to the existence of a reconstruction wavelet.NEWLINENEWLINE The power of the technique is illustrated by numerous applications which extend those of \textit{V. S. Vladimirov} [Methods of the theory of generalized functions, London: Taylor \& Francis (2002; Zbl 1078.46029)] and \textit{V. S. Vladimirov} et al. [Tauberian theorems for generalized functions. Dordrecht etc.: Kluwer Academic Publishers (1988; Zbl 0636.40003)] in Section 7 and by further extensions in Section 8. We also note that the classical Littlewood Tauberian theorem is proved as an easy consequence of the general approach given in the paper.