Approximation in certain intermediate spaces
The author proves the following theorems: Theorem 1: Let \(U_ n\) be an operator of type B, \(g\in B[-1,1]\), \(x\in (-1,1)\) be a point of continuity of g and \(g(x)=0\). Let \(\ell\) be an integer, \(2\leq \ell \leq n- 3\) such that \(x\in (y_{\ell +1,n},y_{\ell,n}]\subseteq [y_{n- 2,n},y_{2,n}]\). Then, \(| U_ n(g,x)| \leq F_{1,n}(x)\sum^{\ell -2}_{k=1}\frac{1}{k}\Omega (g,\frac{k}{\ell - 1},[y_{\ell -1},y_{\ell -1-L(\ell -1)/k}])+\) \[ +F_{2,n}(x)\quad \Omega (g,1,[y_{\ell +2},y_{\ell - 1}])+F_{1,n}(x)\sum^{n}_{k=\ell +2}\frac{1}{k-\ell -1}\times \] \[ \times \Omega (g,\frac{x-\ell -1}{n-\ell},[y_{L(n-\ell)/(x-\ell - 1)]+1},y_{\ell +2}]) \] where \(y_ j\) denotes \(y_{j,n}\) and \[ F_{1,n}(x):=\frac{4C(x)}{D_ 1}(A_ n(x)+D_ 2B_{2,n}(x)) \] and \[ F_{2,n}(x):=3D_ 2B_{1,n}(x)+2A_ n(x)C(x)/D_ 1. \] Theorem 2: Let g,x satisfy the conditions of theorem 1. Suppose that \(| p_ n(t)w(t)\sqrt{1-t^ 2}| \leq c,\) \(t\in [-1,1]\), \(n=0,1,...\). Then \[ w(x)| s_ n(g,x)| \leq c_ 1(x)\sum^{n}_{k=1}\frac{1}{k}\Omega^*(g,\frac{k}{n},[x- c\frac{1+x}{k},x+c\frac{1-x}{k}]) \] where as a function of x, \(c_ 1\) is bounded on every closed subinterval of \((-1,1).\) Theorem 3: Suppose that g,x satisfy the conditions of Theorem 1. We assume that each of the following conditions holds. If \(t\in (-1,1)\) then \(n\lambda_ n(t)\geq W(t)\), \(n=1,2,...\), where W(t) denotes a positive ``constant'' depending on t, but independent of n. \(n\lambda_{kn}\leq c\), \(n\lambda_{kn}p^ 2_{n-1}(x_{kn})\leq c\), \(k=1,...,n\), \(n=1,2,...\), \(c_ 1\leq n(O_{k+1,n}-O_{k,n})\leq c_ 2\), \(k=1,...,n\), \(n=1,2,...\). Then \[ | L_ n(y,x)| \leq c_ 1(x)\{| p_ n(x)| \sum^{n}_{x=1}\frac{1}{x}\Omega^*(g,\frac{x}{n},[x- c\frac{1+x}{n},x+c\frac{1-x}{n}])+ \] \[ +\frac{1}{n}\Omega^*(g,1,[x- c\frac{1+x}{n},x+c\frac{1-x}{n}])\}. \] Theorem 4: Let w be a generalized Jacobi weight, g, x as in Theorem 1. Then \[ | H_ n(g,x)| \leq c_ 1(x)\{p^ 2_ n(x)\sum^{n}_{k=1}\frac{1}{k}\Omega^*(g,\frac{x}{n},[n- c\frac{1+x}{k},n+c\frac{1-x}{k}])+ \] \[ +\frac{1}{n}\Omega^*(g,1,[x- c\frac{1+x}{n},x+c\frac{1-x}{n}])\}. \] The author obtains estimates in the case of a general class of operators. The results are given in terms of an expression, which for continuous functions, is equivalent to the K- functional. As particular cases, the author proves the expansions in certain orthogonal polynomials and Lagrange interpolation at the zeros of orthogonal polynomials and Hermite-Fejer interpolation at the zeros of generalized Jacobi polynomials. The author extends results of \textit{R. Bojanic} [Publ. Inst. Math., Nouv. Ser. 26(40), 57-60 (1979; Zbl 0451.42004), and An estimate of rate of convergence of a general class of orthogonal expansions of functions of bounded variation (manuscript)] and \textit{R. Bojanic} and \textit{F. H. Cheng} [Approximation theory, 2nd Conf. Edmonton/Alberta 1982, CMS Conf. Proc. 3, 5-17 (1983; Zbl 0554.41012)]. This paper covers many important results and is very useful for future workers on this topic.
scientific article; zbMATH DE number 4159349
scientific article