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On approximation of functions by generalized Abel–Poisson operators. (Russian, English) Zbl 0984.42002

Sib. Mat. Zh. 42, No. 4, 926-936 (2001); translation in Sib. Math. J. 42, No. 4, 779-788 (2001).
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., \[ A_{r,l}(f,x)=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x+t)P_{r,l}(t) dt, \quad P_{r,l}(t)=\frac{1}{2}+\sum_{\nu=1}^{\infty}r^{\nu^l}\cos\nu t, \quad 0<r<1,\;l>0. \] The author derives an asymptotic representation for the quantity \[ \Delta_l(r,\alpha)=\sup_{f\in Z_{\alpha}}\|f(x)-A_{r,l}(f,x)\|_{C_{2\pi}}, \] where the Zygmund class \(Z_{\alpha}\) is defined as follows: \[ Z_{\alpha}=\{f\in C_{2\pi}:\;|f(x+h)-2f(x)+f(x-h)|\leq 2|h|^{\alpha}\} \quad (0<\alpha\leq 2,\;|h|\leq 2\pi). \] For example, it is demonstrated that, for \(0<\alpha<l\leq 2\) (\(r\to 1-\)), \[ \Delta_l(r,\alpha)=\frac{2}{\pi}\sin\frac{\alpha\pi}{2}\Gamma(\alpha) \Gamma\Bigl(\frac{l-\alpha}{l}\Bigr)(1-r)^{\alpha/l} + \Lambda(r,\alpha), \] with \(\Gamma(\alpha)\) the gamma-function. The asymptotic behavior of the remainder \(\Lambda(r,\alpha)\) is also described.

MSC:

42A10 Trigonometric approximation
41A25 Rate of convergence, degree of approximation
41A35 Approximation by operators (in particular, by integral operators)
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