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Statistical convergence of multiple sequences. (English) Zbl 1041.40001

In this paper the notion of statistical convergence is generalized from ordinary sequences to double and multiple sequences of real or complex numbers. It is shown in analogy to the ordinary case that statistical convergence is equivalent to a statistical Cauchy property and that a statistical convergent sequence can be split into a convergent sequence and a sequence which vanishes on a set of density one. Furthermore the relation to strong Cesàro summability is considered and as application statistical convergence of Fourier series for functions \(f \in L(\log^+L)^{d-1}([-\pi,\,\pi]^d)\) is investigated.

MSC:

40A05 Convergence and divergence of series and sequences
40B05 Multiple sequences and series
42A24 Summability and absolute summability of Fourier and trigonometric series
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