Tauberian conditions for almost convergence

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 Vallee-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.