On approximation of functions by generalized Abel--Poisson operators (Q2773673)

Let \(C_{2\pi}\) be the class of continuous periodic functions of period \(2\pi\) and let the symbol \(A_{r,l}(f,x)\) denote the generalized Abel-Poisson operator, i.e., NEWLINE\[NEWLINE A_{r,l}(f,x)=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x+t)P_{r,l}(t) dt, \quad P_{r,l}(t)=\frac{1}{2}+\sum_{\nu=1}^{\infty}r^{\nu^l}\cos\nu t, \quad 0<r<1,\;l>0. NEWLINE\]NEWLINE The author derives an asymptotic representation for the quantity NEWLINE\[NEWLINE\Delta_l(r,\alpha)=\sup_{f\in Z_{\alpha}}\|f(x)-A_{r,l}(f,x)\|_{C_{2\pi}},NEWLINE\]NEWLINE where the Zygmund class \(Z_{\alpha}\) is defined as follows: NEWLINE\[NEWLINE Z_{\alpha}=\{f\in C_{2\pi}:\;|f(x+h)-2f(x)+f(x-h)|\leq 2|h|^{\alpha}\} \quad (0<\alpha\leq 2,\;|h|\leq 2\pi). NEWLINE\]NEWLINE For example, it is demonstrated that, for \(0<\alpha<l\leq 2\) (\(r\to 1-\)), NEWLINE\[NEWLINE \Delta_l(r,\alpha)=\frac{2}{\pi}\sin\frac{\alpha\pi}{2}\Gamma(\alpha) \Gamma\Bigl(\frac{l-\alpha}{l}\Bigr)(1-r)^{\alpha/l} + \Lambda(r,\alpha), NEWLINE\]NEWLINE with \(\Gamma(\alpha)\) the gamma-function. The asymptotic behavior of the remainder \(\Lambda(r,\alpha)\) is also described.