Pointwise approximation by algebraic polynomials
The following result is due to \textit{A. F. Timan} [Dokl. Akad. Nauk SSSR 78, 17-20 (1951; Zbl 0042.071)]. For all \(f\in C^ r[-1,1]\), \(r\in {\mathbb{N}}_ 0\) there exists a polynomial \(P_ n\) of degree not greater than n, such that for any \(x\in [-1,1]\) (1) \(| f(x)-P_ n(x)| \leq C\Delta_ n(x)^ r\omega_ 1(f^{(r)},\Delta_ n(x)),\) where \(\Delta_ n(x)=(\sqrt{1-x^ 2}/n)+1/n^ 2\) and \(\omega_ 1(f^{(r)},.)\) is the modulus of continuity of the rth derivative of f. \textit{Yu. A. Brudnyi} [Dokl. Akad. Nauk SSSR 161, 746-749 (1965; Zbl 0132.049)] managed to replace \(\omega_ 1\) in (1) by s-th modulus of continuity \(\omega_ s\), while \textit{I. E. Gopengauz} [Teor. Funkcii, Funkcional'. Analiz. Prilozen. 4, 204-210 (1967; Zbl 0185.136)] proved the estimate simultaneously for derivatives of f: \[ (2)\quad | f^{(k)}(x)-P_ n^{(k)}(x)| \leq C\Delta_ n^{r-k}\omega_ s(f^{(r)},\Delta_ n(x)). \] In 1963, G. G. Lorentz raised the question of whether \(\Delta_ n(x)\) in (1) can be replaced by \(\Gamma_ n(x)=\sqrt{1-x^ 2}/n\). This problem was solved in 1966 by S. A. Telyakowskij. In 1967-1985 \textit{I. E. Gopengauz} [Mat. Zametki 1, 163-172 (1967; Zbl 0165.387)] (case \(s=1)\), \textit{R. De Vore} [Proc. Conf. Approximation of Functions, Kalouga 132-141 (1975)] \((s=1\), \(r=0)\) and \textit{H. Gonska} and \textit{E. Hinnemann} [Approximation Theory IV, Proc. int. Conf., Tex A \& M Univ. 1983, 527-532 (1983; Zbl 0551.41028)] \((s\leq r+2\), \(k=0\) and \(s\leq r\), \(0\leq k\leq r-s)\) proved that in some cases \(\Delta_ n(x)\) can be replaced by \(\Gamma_ n(x)\) in (2) as well. In 1985 \textit{Xiangming Yu} [J. Approximation Theory Appl. 1, No.3, 109-114 (1985; Zbl 0637.41006)] gave a counterexample (for the case \(s=r+3)\) to show that in general it's not possible. The paper establishes that for \(\Delta_ n(x)\) in (2) can be replaced by \(\Gamma_ n(x)\) it's necessary and sufficient that \(0\leq k\leq \min \{r-s+z,r\}\), which solves the problem completely.
scientific article; zbMATH DE number 4148809