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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 \(\mathcal C\).
41A25 Rate of convergence, degree of approximation
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