Multidimensional Tauberian theorems for Banach-space valued generalized functions

Tauberian class estimates for vector-valued distributions

“…The goal of this article is to generalize and improve various of the results of Drozhzhinov and Zav'yalov from [8,9]. The present work might be regarded as a continuation of our previous paper [16], where we treated multidimensional Tauberian theorems for quasiasymptotics of vector-valued distributions (see also [21]).…”

“…The canonical topology of this space is defined in the standard way [11]. We shall also employ an interesting class of subspaces of S(R n ) introduced by Drozhzhinov and Zav'yalov in [9]. Let I be an ideal of the ring C[t 1 , t 2 , .…”

“…We will essentially prove here that the sought optimal Tauberian kernels are given by the class of non-degenerate test functions that we introduced in [16] and turns out to be much larger than the one employed in [8,9] (the latter called in this article the class of strongly non-degenerate test functions, see Definition 5.1). Note that in Wiener Tauberian theory [12] the Tauberian kernels are those whose Fourier transforms do not vanish at any point.…”

Applications of Tauberian theorems in some problems in mathematical physics

“…Theorem 3.1 [12], [13]. Let B and L be Banach spaces, B ⊂ L be linear and continuous, f ∈ L(S → L), and ω(x) ∈ S, and there exist m such that T m e ω (τē) ≡ 0 for anyē, |ē| = 1, and for τ > 0.…”

“…Theorem 3.2 [12], [13]. Let B be a Banach space, ρ(k) be an automodel function of order α, f ∈ L(S → B), and ω(x) ∈ S be such that condition (3.5 N = 0, 1, . .…”