Approximation of functions characterized by one nonsymmetric operator of generalized translation
Let \(L_{p,\alpha,\beta}\) denote the set of all functions for which \(f( x) ( 1-x) ^{\alpha}( 1+x) ^{\beta}\in L_{p},\) \(( 1\leq p\leq\infty) \) and consider the metric \(\|f\|_{p,\alpha,\beta}=\|f( x) ( 1-x) ^{\alpha}( 1+x) ^{\beta}\|_{p}\). Let \(E_{n}( f) _{p,\alpha,\beta}\) be the best approximation of a function \(f\in L_{p,\alpha,\beta}\) by algebraic polynomials of degree \(\leq n-1\) with respect to the metric \(\|\cdot\|_{p,\alpha,\beta}\). The author defines a continuity modulus convenably chosen and denoted by \(\widetilde{\omega }( f,\delta) _{p,\alpha,\beta}\) and proves that if \(p=1\) and \(\alpha\in (0,1/2] \), or \(p\in( 1,\infty) \) and \(\alpha\in( 1/2 - 1/(2p), 1-1/(2p)) \), or \(p=+\infty\) and \(\alpha\in[ 1/2,1] \), then for every \(f\in L_{p,\alpha+1,\alpha}\) and every \(n\in\mathbb{N}\) the following inequality holds: NEWLINE\[NEWLINE C_{1}\cdot E_{n}( f) _{p,\alpha+1,\alpha}\leq\widetilde{\omega }( f;{\textstyle\frac{1}{n}}) _{p,\alpha+1,\alpha}\leq C_{2} {\textstyle\frac{1}{n^{2}} }\sum\limits_{\nu=1}^{n}\nu E_{\nu}( f) _{p,\alpha+1,\alpha} NEWLINE\]NEWLINE where \(C_{1},C_{2}>0\) are independent of \(f\) and \(n\).NEWLINENEWLINEFor the entire collection see [Zbl 0952.00006].
scientific article; zbMATH DE number 1607203