Absolute matrix summability of the allied series of a Fourier series
Let \(A=(a_{nk})\) be a real infinite matrix such that \(a_{nk}=0\) \((k>n)\), \(\lim_{n\to \infty}a_{nk}=0\), and \(a_{n-1,k}\geq a_{n,k+1}\geq a_{nk}\) for all \(k\leq n-1\); (these imply that \(a_{nk}\geq 0\) for all n,k). Assume also that \[ (n+1)(a_{n-1,k}- a_{n,k+1})\geq (n-k)a_{nk}-(n-k-1)a_{n,k+1}, \] \(\sum^{n}_{k=0}a_{nk}=1\) \((n=0,1,...)\), \(\sum^{\infty}_{n=0}a_{no}/(n+1)<\infty\), and that for a given \(\alpha\in (0,1)\), \(\sup_{k}k^{- \alpha}\sum^{\infty}_{n=k}n^{\alpha -1}\sum^{n}_{r=n- k}a_{nr}<\infty.\) Let f be a \(2\pi\)-periodic function, L-integrable on \((-\pi,\pi)\), and denote \[ \psi(t):= \{f(x+t)-f(x-t)\},\quad \Psi_{\alpha}(t):=\frac{1}{\Gamma (\alpha)}\int^{t}_{0}(t- u)^{\alpha -1}\psi (u)du\;(\alpha >0,\quad t>0). \] Under the above conditions on the matrix A, it is proved that if \(\psi_{\alpha}(+0)=0\) and \(\int^{\pi}_{0}t^{-\alpha}| d\Psi_{\alpha}(t)| <\infty\), then the conjugate Fourier series of f is absolutely summable \(| A|\) at the point x. Specializations to Nörlund and Cesàro matrices lead to known results.
scientific article; zbMATH DE number 4157207