Tauberian Theorems for Sequences linked by a Convolution

Abstract

Given sequences 9 , X : IN0 + C , g(0) = 1, linked by the convolution X * g = 9'(g'(n) := (n + 1)g(n + 1)) we study what can be inferred about X (n --f m) from some concrete information about the behaviour of g (n + 00).

Tauberian Theorems for Sequences Satisfying Certain Convolution Relations

“…As we have seen at the beginning of the preceding section, a quantitative asymptotic formula for λ like in Theorem 2.11 from [5] cannot be proved in the same way as our Theorem 1, because Wiener's inversion theorem does not give any quantitative estimate for the remainder m≥n h −1 (m) . An estimate of this kind can be found using results of Lucht and Reifenrath [4] generalizing Wiener's inversion theorem.…”

“…The very special case k = 1 corresponds to (1.2). If the growth condition r ∈ 1 is strengthened to r(n) n −γ with some γ > 3, and the further assumption λ(n) ≥ 0 for all n is made, Theorem 2.11 from [5] asserts that…”

“…Under the conditions λ(n) ≥ 0 for all n and γ > 3/2 , Proposition 2.3 and Corollary 2.4 from [5] guarantee that G(z) is zero-free in U with the possible exception of a zero at z = −1. Therefore we have …”

“…with constants A > 0, q > 1 and remainder terms either r(n) = O(s n ) with some positive constant s < q Knopfmacher's Axiom A # , see [3] , or r(n) = O(n −γ q n ) with some constant γ > 1 (see Zhang [6]), or r(n) = O n −1 q n as n → ∞ (see Warlimont [5]). These conditions together with suitable assumptions concerning poles and zeros of G(z) for |z| < 1/q lead to estimates for λ(n), from which the Abstract Prime Number Theorem π(n) = 1 n q n + (n) with certain estimates for the remainder term (n) can be derived by elementary arguments.…”

“…A crucial technical point, due to the origin of the problem, is that λ ≥ 0. Warlimont [5] pointed out that the asymptotic evaluation of the sequence λ from the above assumptions concerning the sequence g can be considered from the view of summability, i. e., the problem consists in proving Tauberian theorems on λ from certain assumptions concerning g under the condition λ ≥ 0. Dividing g(n) and π(n) by q n transforms the disc |z| < 1/q to the open unit disc U in the complex plane, and leads to the normalization q = 1.…”