The effects of matrix summability methods on bounds of function sequences. (Q1767212)

Let \(A\) be an infinite matrix, \(\mathcal{F}\) a space of function sequences, \(A\mathcal{F} := \sum_{k = 1}^{\infty}a_{nk}f_k(x)\). The symbol \(G(\mathcal{F}; D)\) denotes the least upper bound of the element \(f_n(x)\) over the subset \(D\). \textit{R. P. Agnew} [Trans. Am. Math. Soc. 32, 669--708 (1930; JFM 56.0213.01)] obtained necessary and sufficient conditions on \(A\) for \(\lim \inf G(\mathcal{F}; D) \leq \lim \inf G(A\mathcal{F}; D) \leq \lim \sup G(A\mathcal{F}; D) \leq \lim \sup G(\mathcal{F}; D)\). In this paper the author obtains the corresponding results with the outsides of this inequality replaced by \(\lim \inf\) and \(\lim \sup\) of the statistical limit of a regular matrix transformation \(T\) of \(G(\mathcal{F}; D)\), respectively.