A Wiener Tauberian theorem for operators and functions

Abstract In this paper we generalize some classical Tauberian theorems for single sequences to double sequences. One-sided Tauberian theorem and generalized Littlewood theorem for (C; 1; 1) summability method are given as corollaries of the main results. Mathematics Subject Classification 2010: 40E05, 40G0

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A genuine analogue of the Wiener Tauberian theorem for some Lorentz spaces on SL(2,ℝ)

Forum Mathematicum ◽

10.1515/forum-2020-0134 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Tapendu Rana

Keyword(s):

Tauberian Theorem ◽

Lorentz Spaces

AbstractIn this paper, we prove a genuine analogue of the Wiener Tauberian theorem for {L^{p,1}(G)} ({1\leq p<2}), with {G=\mathrm{SL}(2,\mathbb{R})}.

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GENERALIZATION OF THE EFFECTIVE WIENER–IKEHARA THEOREM

International Journal of Number Theory ◽

10.1142/s1793042113500760 ◽

2013 ◽

Vol 09 (08) ◽

pp. 2091-2128 ◽

Cited By ~ 3

Author(s):

SZILÁRD GY. RÉVÉSZ ◽

ANNE de ROTON

Keyword(s):

Laplace Transform ◽

Tauberian Theorem ◽

Error Term ◽

Effective Version ◽

Slow Decrease ◽

Asymptotic Evaluation ◽

The Laplace Transform

We consider the classical Wiener–Ikehara Tauberian theorem, with a generalized condition of slow decrease and some additional poles on the boundary of convergence of the Laplace transform. In this generality, we prove the otherwise known asymptotic evaluation of the transformed function, when the usual conditions of the Wiener–Ikehara theorem hold. However, our version also provides an effective error term, not known thus far in this generality. The crux of the proof is a proper, asymptotic variation of the lemmas of Ganelius and Tenenbaum, also constructed for the sake of an effective version of the Wiener–Ikehara theorem.

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The convolution algebraH1(R)

Journal of Function Spaces and Applications ◽

10.1155/2010/524036 ◽

2010 ◽

Vol 8 (2) ◽

pp. 167-179 ◽

Cited By ~ 3

Author(s):

R. L. Johnson ◽

C. R. Warner

Keyword(s):

Banach Algebra ◽

Maximal Ideal ◽

Tauberian Theorem ◽

Singular Integrals ◽

Approximate Identity ◽

Maximal Ideal Space ◽

Ideal Space

H1(R) is a Banach algebra which has better mapping properties under singular integrals thanL1(R) . We show that its approximate identity sequences are unbounded by constructing one unbounded approximate identity sequence {vn}. We introduce a Banach algebraQthat properly lies betweenH1andL1, and use it to show thatc(1 + lnn) ≤ ||vn||H1≤Cn1/2. We identify the maximal ideal space ofH1and give the appropriate version of Wiener's Tauberian theorem.

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