Potapov, M. K.; Berisha, F. M. Direct and inverse theorems of approximation theory for a generalized modulus of smoothness. (English) Zbl 0957.41009 Anal. Math. 25, No. 3, 187-203 (1999). 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 \(\mathcal C\). Reviewer: Ion Badea (Craïova) MSC: 41A25 Rate of convergence, degree of approximation Keywords:operator of generalized translation; generalized modulus of smoothess; direct and inverse theorems of approximation PDF BibTeX XML Cite \textit{M. K. Potapov} and \textit{F. M. Berisha}, Anal. Math. 25, No. 3, 187--203 (1999; Zbl 0957.41009) Full Text: DOI arXiv References: [1] Беитмен, Г.; Ердеии, А., Высщие трансцендентные функции.2 (1965), Москва: Фиэматгиэ, Москва [2] Butzer, P. L.; Stens, R. L.; Wehrens, M., Higher order of continuity based on the Jacobi translation operator and best approximation, C. R. Math. Rep. Acad. Sci. Canada, 2, 83-87 (1980) · Zbl 0436.41013 [3] Pawelke, S., Ein Satz von Jackonsonschen Typ für algebraische Polynome, Acta Sci. Math. (Szeged), 33, 323-336 (1972) · Zbl 0243.41005 [4] Потапов, М.К., О приближении алгебраическими многочленами в интегральной метрике с весом Якоби, Вестник МГУ, серия 1, матем. и мех., 4, 43-52 (1983) [5] Потапов, М. К., Некоторые неравенства для полиномов и их проиэводных, Вестник МГУ серия 1, mamem. u mex., 2, 10-20 (1960) [6] Потапов, М. К., О структурных и конструктивных характеристиках некоторых классов функций, Труды МИАН, 131, 211-231 (1974) · Zbl 0314.41026 [7] Potapov, M. K.; Berisha, F. M., On coincidence of classes of functions defined by the generalised modulus of smoothness of orderk or by the orrder of the best approximation by algebraic polynomials, East J. Approx., 4, 2, 217-241 (1998) · Zbl 0932.41006 [8] Потапов, М. К.; Федоров, В. М., О теоремах Джексона для обобшенного модуля гладкости, Труды МИАН, 172, 291-295 (1985) [9] Виленкин, Н. Я., Специальные функции и теория представлении групп (1965), Москва: Наука, Москва · Zbl 0144.38003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.