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On the connection between an \(r\)-order modulus of smoothness and the best approximation by algebraic polynomials. (Russian. English summary) Zbl 0960.41004
Let \(1\leq p<\infty\) and \(L_{p}=L_{p}[ -1,1] .\) One denotes by \(L_{p,\alpha}\) the space of all functions \(f\) such that \(f( x) ( 1-x^{2}) ^{\alpha}\in L_{p}\), equipped with the norm \(\|f\|_{p,\alpha}=\|f( x) ( 1-x^{2}) ^{\alpha}\|_{L_{p}}\). Let \(E_{n}( f) _{p,\alpha} =\inf_{P_{n}}\|f-P_{n}\|_{p,\alpha}\) be the best approximation of \(f\) by algebraic polynomials of degree \(\leq n-1.\) By introducing a non-symmetric generalized translation operator, the authors define a generalized smoothness modulus of order \(r\), denoted by \(\widehat{\omega} _{r}( f,\delta) _{p,\alpha}.\) Let \(p,\alpha\) and \(r\) be given such that \(1\leq p<\infty,\) \(r\in\mathbb{N}\), \(\frac{1}{2}<\alpha\leq 1\) for \(p=1\); \(1-\frac{1}{2}<\alpha<\frac{3}{2} -\frac{1}{2p}\) for \(1<p<\infty\); \(1\leq\alpha<\frac{3}{2}\) for \(p=+\infty\), and let \(f\in L_{p,\alpha}\). Under these hypotheses one proves that a) for every \(\delta\in[ 0,\pi] \) the inequalities \[ C_{1}( \cos^{4}\delta/2) ^{r( r-1) }K_{r}( f,\delta) _{p,\alpha}\leq\widehat{\omega}_{r}( f,\delta) _{p,\alpha}\leq C_{2}\frac{1}{( \cos^{4}\delta/2) ^{r}} K_{r}( f,\delta) _{p,\alpha}, \] hold, where \(K_{r}( f,\delta) _{p,d}\) is Peetre’s functional and \(C_{1},C_{2}\) are independent of \(f\) and \(\delta.\) b) for every \(n\in\mathbb{N}\) \[ C_{1}E_{n}( f) _{p,\alpha}\leq\widehat{\omega}_{r}( f,1/n) _{p,\alpha}\leq C_{2}\frac{1}{n^{2r}} {\textstyle\sum\limits_{\nu=1}^{n}} \nu^{2r-1}E_{\nu}( f) _{p,\alpha}, \] where \(C_{1}\) and \(C_{2}\) does not depend on \(f\) and \(n.\)
MSC:
41A10 Approximation by polynomials
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
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