A Jackson-type estimation on the real line by interpolation (Q1099342)

In 1967, O. Kis and P. Vertesi introduced a new interpolation process on equidistant nodes \(\{t_{kn}\}\), \(t_{kn}=2k\pi /(2n+1)\), \((k=0,\pm 1,\pm 2,...\); \(n=0,1,2,...)\) to prove the Jackson theorem for \(f\in C_{2\pi}\) (the class of \(2\pi\)-periodic continuous functions). An extension of this result was given by \textit{R. Gervais, Q. I. Rahman} and \textit{G. Schmeiser} [Polynomial and Spline Approximation Proc. NATO adv. Study Inst., Calgary 1978, NATO adv. Study Inst. Ser., Ser. C, math. phys. Sci. 49, 203-223 (1979; Zbl 0407.41009)]. In the present paper the author defines a new interpolation process for \(f\in UCB(R)\) (uniformly continuous and bounded). For \(\sigma >0\) and \(R\in {\mathbb{Z}}\), set \[ q_{k\sigma}(z)=\frac{\sin (\sigma (z-x_{k\sigma}))}{\sigma (z- x_{k\sigma})},\quad x_{k\sigma}=\frac{k\pi}{\sigma}+\alpha_{\sigma},\quad u_{k\sigma}(z)=4q^ 3_{k\sigma}(z)-3q^ 4_{k\sigma}(z), \] \[ U_{\sigma}(f,Z)=\sum^{\infty}_{k=- \infty}f(x_{k\sigma})u_{k\sigma}(z),\quad f\in U\subset B(R). \] The main theorem asserts that for arbitrary \(f\in UCB(R)\), \[ \sup_{x\in R}| f(x)-U_{\sigma}(f,x)| =O(\omega (f,\frac{1}{6})),\quad \sigma >0. \] Applications of the above result are given to obtain analogues of two theorems of Gervais, Rahman and Schmeiser.