On absolute matrix summability factors of infinite series and Fourier series (Q2037513)

The author generalizes a result of \textit{H. Bor} [Bull. Sci. Math. 169, Article ID 102990, 8~p. (2021; Zbl 1465.42005)]. Let \(\sum a_n\) be a given infinite series and \((\lambda_n)\) an arbitrary real sequence. The new theorem reads as follows. Let \((X_n)\) be an almost increasing sequence and \(A = (a_{nv})\) be a positive normal matrix. Let $(\overline{a}_{nv})$ be the matrix of the series-to-sequence transformation and $(\hat{a}_{nv})$ be the matrix of the series-to-series transformation. Assume \[ \overline{a}_{n0}=1,\quad n=0,1,2,\dots , \] \[ {a}_{n-1,v}\geq {a}_{nv}\quad \text{for}\quad n\geq v+1, \] \[ {a}_{nn}=\mathcal{O}\left(\frac{p_n}{P_n}\right), \] \[ \sum_{v=1}^{n-1}\frac{\hat{a}_{n,v+1}}{v}=\mathcal{O}\left({a}_{nn}\right), \] and \[ \sum_{n=1}^{m}{a}_{nn}\frac{|t_n|^k}{X_n^{k-1}}=\mathcal{O}\left( X_{m}\right)\quad \text{as}\quad m\to \infty . \] If the conditions (1)--(3) of Bor's theorem are satisfied, then the series \(\sum a_n\lambda _n\) is summable \(|A, p_n|_k\), \(k \geq 1\). Finally, an application of this result to trigonometric Fourier series is given as well.