On the generalized Cesàro summability factors.
Recently, Tripathy - Jour. Ind. Math. Soc., 32 (1960), 141-154 - has proved some results on absolute Hausdorff summability of some series associated with Fourier series and its allied series, which generalise the results proved by Mohanty on absolute Cesaro summability. Proceeding on the similar lines, the author has generalised the results of Cheng - Duke Math. Jour., 15 (1948), 17-27 - by proving them on absolute Hausdorff summability.
For any Borel ideal ℐ we describe the ℐ-Baire system generated by the family of quasi-continuous real-valued functions. We characterize the Borel ideals ℐ for which the ideal and ordinary Baire systems coincide.
In the paper, some kind of independence between upper metric dimension and natural order of converging sequences is shown — for any sequence converging to zero there is a greater sequence with an arbitrary () upper dimension. On the other hand there is a relationship to summability of series — the set of elements of any positive summable series must have metric dimension less than or equal to .
Estimates of the strong means of Marcinkiewicz type with the Cesaro means of negative order in one of the variables instead of square partial sums are obtained by characteristics constructed on the basis of moduli of continuity.
If is a sequence of real numbers which is good for the ergodic theorem, is the sequence of the integer parts good for the ergodic theorem ? The answer is negative for the mean ergodic theorem and affirmative for the pointwise ergodic theorem.
In [11] and [7], the concepts of σ-core and statistical core of a bounded number sequence x have been introduced and also some inequalities which are analogues of Knopp’s core theorem have been proved. In this paper, we characterize the matrices of the class and determine necessary and sufficient conditions for a matrix A to satisfy σ-core(Ax) ⊆ st-core(x) for all x ∈ m.