Approximation by convex sums of convolution type operators in Banach spaces (Q2720305)

Let \(X\), \(B(X)\) denote a Banach space and the algebra of all bounded linear \(X\to X\) operators, and let \(\{ T_t:t\in {\mathbb R}\}\) be a \(2\pi\)-periodic, strongly continuous group of operators in \(B(X)\). Given some \(W \in B(X)\) and \(\chi \in L_{2\pi}^1\), the author considers the operator NEWLINE\[NEWLINE(\chi *W)(f)={1 \over 2\pi}\int_{-\pi}^{\pi}\chi(t)T_t(W(f)) dt.NEWLINE\]NEWLINE Many standard approximation operators can be represented in this form if one sets \(W\) to be the Fourier sum and and \(T_t\) to be the translation \(f(\cdot) \to f(\cdot -t)\). Under some additional assumptions concerning \(W\) and \(\chi\), the author finds estimates for \(\|(\chi *W)(f)-Wf\|\) and \(\|(\chi *W)(f)-f \|\), with an emphasis on certain operators \(\chi *W\) generalizing the classic Rogosinski operators.