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Lacunary statistical summability. (English) Zbl 0786.40004
Let θ=(k r ) be an increasing sequence of integers such that k 0 =0, h r :=k r -k r-1 as r, and I r :=(k r-1 ,k r ]. A complex-valued sequence x=(x k ) is said to be S θ -convergent to L if, for each ε>0, we have lim r h r -1 |{kI r : |x k -L|ε}|=0, where |{·}| denotes the cardinality of the set; we then write x k L(S θ ). Likewise, x is an S θ -Cauchy sequence if there is a subsequence (x k ' (r) ) with k ' (r)I r for each r, lim r x k ' (r) =L, and for each ε>0, lim r h r -1 |{kI r : |x k -x k ' (r) |ε}|=0. It is first shown (Theorem 2) that x is S θ -convergent if and only if x is an S θ -Cauchy sequence. Further (Theorem 4) if x is a bounded sequence and x k L(S θ ) then x k L(C 1 ); that is, l S θ C 1 . On the other hand (Theorem 6), if x is unrestricted, then no matrix summability method can include S θ . Finally, let T θ denote the class of non-negative summability matrices A=(a nk ) such that (a) k=1 a nk =1 for every n, and (b) if K with lim r h r -1 |KI r |=0 then lim n k a nk =0. It is shown (Theorem 9) that xl S θ if and only if x is A-summable for every AT θ .
MSC:
40G99Special methods of summability
40A05Convergence and divergence of series and sequences
40D20Summability and bounded fields of methods
40C05Matrix methods in summability