## Direct and inverse theorems of approximation theory for a generalized modulus of smoothness.(English)Zbl 0957.41009

By $$L_p$$ the authors denote the set of functions $$f$$ such that, in the case $$1\leq p<\infty$$, $$f$$ is measurable on $$[-1,1]$$ and $$\|f\|_p=\left(\int^1_{-1}|f(x)|^p dx\right)^{\frac 1p} <\infty$$. In the case $$p=\infty$$, $$f$$ is continuous on $$[-1,1]$$ and $$\|f\|_\infty=\max\{|f(x)|:x\in [-1,1]\}$$. Denote by $$L_{p,\alpha}$$ the set of functions $$f$$ such that $$f(x)(1-x^2)^{\alpha}\in L_p$$ and put $$\|f\|_{p,\alpha}= \|f(x)(1-x^2)^\alpha\|_p$$. For $$f\in L_{p,\alpha}$$, $$E_n(f)_{p,\alpha}=\inf\{\|f-p_n\|_{p,\alpha}:p_n\in\mathcal P_n\}$$ where $$\mathcal P_n$$ is the set of algebraic polynomials of degree not greater than $$n-1$$.
The authors define the operator of generalized translation $\mathcal C(f,t,x)= \pi^{-1}(1-x^2)^{-1}\cos^{-4}(2^{-1} t) \int^\pi_0 (2A^2(x,t,\varphi)-1+B^2(x,t,\varphi))(f\circ B)(x,t,\varphi)d\varphi,$ where $A(x,t,\varphi)=\sqrt{1-x^2}\cos t-x\sin t \cos \varphi +\sqrt{1-x^2}(1-\cos t)\sin t \cos \varphi$ and $$B(x,t,\varphi)=x\cos t +\sqrt{1-x^2}\sin t \cos \varphi$$.
For $$f\in L_{p,\alpha}$$ define the generalized modulus of smoothness $\widehat \omega(f;\delta)_{p,\alpha}=\sup\{\|\mathcal C(f,t,x)-f(x)\|_{p,\alpha}: |t|\leq\delta\}.$ The authors prove direct and inverse theorems of approximation of the operator $$\mathcal C$$.

### MSC:

 41A25 Rate of convergence, degree of approximation
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### References:

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