On strong convergence of trigonometric and Fourier series (Q796019)

A real or complex valued sequence \((s_ k)\) is (i) strongly \(C_ 1- summable\) to t with index \(\lambda>0\) (or \((s_ k)\to t[C_ 1]_{\lambda})\) if \(\sum^{n}_{k=0}| s_ k- t|^{\lambda}=o(n) (n\to \infty)\) and (ii) strongly convergent to t with index \(\lambda>0\) (or \((s_ k)\to t[I]_{\lambda})\) if and only if \((s_ n)\) converges to t and \(\sum^{n}_{k=0}k^{\lambda}| s_ k-s_{k-1}|^{\lambda}=o(n) (n\to \infty)\). In this paper the author proves seven theorems; three of them are concerned with trigonometric series. In the first two theorems he proves that if, for \(\lambda \geq 1\), \({1\over2}a_ 0+\sum^{\infty}_{k=1}(a_ k\quad \cos kx+b_ k\quad \sin kx)\) is \([I]_{\lambda}-convergent\) for either x belonging to a set of positive measure or on a set of second category then the sequences \((na_ n)\) and \((nb_ n)\) are \([C_ 1]_{\lambda}-\) summable to zero. The last four theorems are concerned with the Fourier series of continuous functions. One of his results reads: If f is a function of bounded variation with period 2\(\pi\), then the Fourier series is [I]-convergent uniformly to f if and only if f is continuous with period 2\(\pi\).