Tauberian conditions for almost convergence

Tauberian conditions for almost convergence In [Acta Math. 80(1948), 167–190], G. G. Lorentz characterized almost convergent sequences in $$\mathbb{R}$$ (or in $$\mathbb{C}$$ ) in terms of the concept of uniform convergence of the de la Vallée-Poussin means. In this paper, we present Tauberian results which relate almost convergence to norm convergence or to the (C, 1) convergence. Our results generalize Kronecker lemma. As a consequence, we prove that almost convergence and norm convergence are equivalent for the sequence of the partial sums of the Fourier series of $$f \in L^p(T)$$ (or $$f \in C(T)$$ ), where $$1\leq p \leq \infty$$ . We also show that our results can be used to derive Fatou’s theorem. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Tauberian conditions for almost convergence

Positivity , Volume 13 (4) – Nov 24, 2008
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Publisher
Birkhäuser-Verlag
Copyright
Copyright © 2008 by Birkhäuser Verlag Basel/Switzerland
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-008-2282-z
Publisher site
See Article on Publisher Site

Abstract

In [Acta Math. 80(1948), 167–190], G. G. Lorentz characterized almost convergent sequences in $$\mathbb{R}$$ (or in $$\mathbb{C}$$ ) in terms of the concept of uniform convergence of the de la Vallée-Poussin means. In this paper, we present Tauberian results which relate almost convergence to norm convergence or to the (C, 1) convergence. Our results generalize Kronecker lemma. As a consequence, we prove that almost convergence and norm convergence are equivalent for the sequence of the partial sums of the Fourier series of $$f \in L^p(T)$$ (or $$f \in C(T)$$ ), where $$1\leq p \leq \infty$$ . We also show that our results can be used to derive Fatou’s theorem.

Journal

PositivitySpringer Journals

Published: Nov 24, 2008

References

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