On multipliers for the absolute matrix summability (Q5916314)

Let \((a_{n, k})\) be a triangular matrix with \(a_{n,k}\geq 0\), \(\sum^ n_{r= 0} a_{n, r}= 1\), \(a_{n- 1,k}\geq (1+ a_{n, 0}) a_{n,k+ 1}\), \(0\leq k< n\), \(n\geq 1\) and satisfying \(\sum^ \infty_{n= k} {A_{n,n- k}\over n+ 1}\leq C\) for every \(k\), \(\sum^{n- 1}_{k= 1} | \Delta_ k a_{n, k}|= O(1/n)\), \(\sum^{n- 2}_{k= 0} (k+ 1)| \Delta^ 2_ k a_{n,k}|= O(1/n)\), \(V_{n, k}\leq C\), uniformly for \(0\leq k\leq n\), \(\sum^ \infty_{n= k} {1\over n+ 1} \sum^{n- 1}_{k= 0} | \Delta_ k V_{n, k}| A_{n, k+1}< \infty\), where \(A_{n, k}= \sum^ n_{r= k} a_{n, r}\) and \(V_{n, k}= {(n- k+ 1) a_{n, k}\over A_{n, k}}\). Under these conditions, the authors prove that \(\sum^ \infty_ 1 \varepsilon_ n A_ n(x)\) is summable \(| T|\) whenever \(\varphi_ 1(t)\in \text{BV}(0, \pi)\) provided \(\{\varepsilon_ n\}\) and \(\{\Delta \varepsilon_ n\}\) are of bounded variation, where \(\sum A_ n(x)\) denotes the Fourier series of a \(2\pi\)-periodic Lebesgue integrable function \(f(t)\), at \(t= x\) and \(\varphi_ 1(t)= {1\over t} \int^ t_ 0 \varphi(u)du\), \(\varphi(u)= {1\over 2}\{f(x+ u)+ f(x- u)\}\). This result extends a theorem of \textit{H. P. Dikshit} [Proc. Camb. Philos. Soc. 65, 495-505 (1969; Zbl 0169.399)].