On Grünwald interpolation polynomial operators and approximation of unbounded functions (Q2735374)

A Grünwald interpolation polynomial operator is defined as NEWLINE\[NEWLINEL_n\bigl[ f(t);x\bigr]= \sum^n_{k=1} f(x_k^{(n)}) \bigl[l_k^{(n)}(x) \bigr]^2,NEWLINE\]NEWLINE where \(l_k^{(n)} (x)=\omega_n (x)/[\omega_n (x_k^{(n)})(x-x_k^{(n) })]\), \(\omega_n(x) =\prod^n_{k=1} (x-x_k^{(n)}), -1<x_1^{(n)}< \cdots<x_n^{(n)} <1\) are the zeros of \(n\) Jacobi polynomials. The main theorems areNEWLINENEWLINENEWLINE1. If \(f(x) \in C[-1,1]\), then \(\lim_{n\to \infty} L_n[f(t); x]=f(x)\) almost uniformly in \((-1,1)\); NEWLINENEWLINENEWLINE2. If \(\alpha_n \uparrow\infty\), \(\beta_n \downarrow 0\), and \(\alpha_n \beta_n\to 0,\) \(\Omega(\alpha_n)/(n \beta_n)^2\to 0\), \((n\to\infty)\), then for \(\forall f(x)\in C(\Omega)\), we have \(\lim_{n\to \infty} L_n[f(\alpha_n t); \alpha_n^{-1}x] =f(x)\), and almost uniformly in \((-\infty, \infty)\). And there are 3 corollaries in the paper.