Generalization of the Rogosinski-Bernstein trigonometric summability methods (Q909161)

A series \(\sum^{\infty}_{k=0}a_ k\) of reals or complex numbers with partial sums \(S_ n=\sum^{n}_{k=0}a_ k\) is called summable to t by the Rogosinski-Bernstein method \((B_{h,r})\) of orders h, r if \(\lim_{n\to \infty}\sum^{n}_{k=0}a_ k\cos \frac{(k+r)\pi}{2(n+h)}=t.\) A series is summable to \(t^*\) by the method (C,1), when \(\lim_{h\to \infty}\frac{1}{n+1}\sum^{n}_{k=0}S_ k=t^*.\) The following results are given. 1. For any h and r (C,1) summability implies the \((B_{h,r})\) summability. 2. If h-r\(\leq 1/2\), \(h\neq r\), then the method \((B_{h,r})\) is stronger than (C,1). 3. If h- r\(\in (,1)\) then \((B_{h,r})\) and (C,1) are equivalent. In the case \(r=0\) these imply the results by \textit{R. P. Agnew} [Ann. Math. II. Ser. 537-59 (1959; Zbl 0048.041); \textit{G. M. Petersen}, Pacif. J. Math. 4, 73- 77 (1954; Zbl 0056.058)].