## 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)

### Keywords:

generalized modulus of smoothness; best approximation
Full Text: