A TAUBERIAN THEOREM FOR UNIFORMLY WEAKLY CONVERGENCE AND ITS APPLICATION TO FOURIER SERIES

Abstract

In 1995, S. Mercourakis introduced the concept of uniformly weakly convergent sequences and characterized such sequences as those with the property that any of its subsequences is Ces`aro-summable. In this paper, we present a Tauberian theorem for such kind of convergence. As a consequence, we prove that the uniformly pointwise convergence and the uniform convergence of a sequence of complex-valued functions coincide under a suitable Tauberian condition. This result affirmatively answers a question raised by S. Mercourakis concerning the Fourier series of a continuous function on the circle group T. In this paper, a result of Banach type is also established for uniformly weakly convergent sequences. Our result generalizes the work of Mercourakis.