STATISTICAL AND LACUNARY CONVERGENCE-ITS LATEST DEVELOPMENT FROM THE POINT OF VIEW OF SEQUENCE SPACES

Abstract

The development of the theory of sequence spaces is nowadays effected by the introduction of new convergence methods and theories in the process. Some of them are statistical convergence, lacunary convergence, lacunary statistical convergence etc (see [1], [2], [3], [8], [11]). Here, I have basically concentrated on these concepts. Statistical convergence while introduced over nearly fifty years ago has only recently become an area of active research in sequence spaces. Fast [5] extended the concept of sequential limit which he called statistical convergence. Schoenberg (1959) gave some basic properties of statistical convergence and studied the concept as summability method. The basic concept of statistical convergence is based on the notion of natural density of sets A ⊆ N = {1, 2, . . . , n, . . .}, refer to [5], [6]. If A ⊆ N and A(n)=|A ∩ {1, 2, . . . , n}|, where the vertical bar denotes the cardinality of the