Saturation classes of linear methods of summation of Fourier series (Q805969)

Let X be one of the Banach spaces C or \(L_ p\), \(1\leq p\leq \infty\), of continuous or integrable, \(2\pi\)-periodic functions with the usual norms and \(\hat X=\{\hat f:\;f\in X\},\) where \(\hat f=(\hat f(k))\) is the sequence of the Fourier coefficients of f. For a numerical sequence \(\psi =(\psi (k))^{\infty}_{k=1},\quad \psi (k)\neq 0,\) let \(X^{\psi}_ 0\) denote one of the spaces \(\{f\in C:\;(\hat f(k)/\psi (| k|))_{k\neq 0}\in \hat L_{\infty}\}\) or \(\{f\in L_ p:\;(\hat f(k)/\psi (| k|))_{k\neq 0}\in \hat L_ p\}.\) Given a triangular matrix method \(\lambda =[\lambda_ k^{(n)}],\) let \(U_ n(\lambda),\quad n=1,2,...,\) be the linear operators from X into X defined by \(U_ n(\lambda)f(x)=\sum_{| k| \leq n- 1}\lambda^{(n)}_{| k|} \hat f(k)e^{ikx}.\) A method \(\lambda\) is saturated in X if there exists a sequence \(\phi (n)\downarrow 0\quad (n\to \infty)\) such that \(\| f-U_ n(\lambda)f\|_ X=o(\phi (n))\quad (n\to \infty)\) implies \(f=const.\) and \((1)\quad \| f-U_ n(\lambda)f\|_ X=O(\phi (n))\quad (n\to \infty)\) for some \(f\in X.\) The set F(X,\(\lambda\)) of all \(f\in X\) for which (1) holds is called the saturation class of \(\lambda\). If there exist sequences \(\phi (n)\downarrow 0\quad (n\to \infty)\) and \(\psi (k)\neq 0\) such that \((2)\quad \lim_{n\to \infty}(1-\lambda_ k^{(n)})/\phi (n)=1/\psi (k),\quad k=1,2,...,\) then it is well known that \(\lambda\) is saturated in X and that \(F(X,\lambda)\subset X^{\psi}_ 0.\) The author considers saturation classes of the methods \(\lambda\) satisfying (2) for which \((\| U_ n(\lambda)\|)^{\infty}_{n=1}\) is not necessarily bounded, interpreting \(\| \|\) as the norm of the operators \(U_ n(\lambda)\) from C into C. He proves (Theorem 1) that under these assumptions \(X_ 0^{\psi^*}\subset F(X,\lambda)\subset X^{\psi}_ 0\) provided that the sequence \(\gamma^{(n)}_{| k|}=(1-\lambda_ k^{(n)})\psi^*(| k|)/\phi (n),\quad k\neq 0,\quad \gamma_ 0^{(n)}=1,\) where \(\psi^*(| k|)\leq \psi (| k|),\) generates uniformly bounded multipliers \(\gamma^{(n)}\) of the type \((L_{\infty},L_{\infty})\) in the case \(X=C,\) and of the type \((L_ p,L_ p),\quad 1<p<\infty,\) in the case \(X=L_ p\) and gives examples of such methods \(\lambda\) for which \(F(X,\lambda)\neq X^{\psi}_ 0.\) The idea of the proof of this result is contained in a book by \textit{P. L. Butzer} and \textit{R. J. Nessel} [Fourier analysis and approximation, Vol. 1 (1971; Zbl 0217.426)].