Regular statistical convergence of multiple sequences
Abstract The concepts of statistical convergence of single and multiple sequences of complex numbers were introduced in [2] and [8], respectively. In this paper, we introduce the concept indicated in the title. We prove that if a d-multiple sequence is regularly statistically convergent, then its statistical limit can be computed as the iterated statistical limit of the statistical limits of its subsequences which correspond to an arbitrary partition of the index set {1,2,..., d}, d ≥ 2. As an application, we prove that if a function ƒ is in Lp(Td) for some p > 1, T = [0, 2π), then the (symmetric) rectangular partial sums of its d-multiple Fourier series are regularly statistically convergent to ƒ { u1, . . . , ud) at almost every point (u1,..., ud) ∈Td. Furthermore, if ƒ is in C(Td), then the regular statistical convergence of the rectangular partial sums takes place uniformly on Td.