Interpolation of functions on the real axis and applications (Q2703343)

The paper deals with the approximate properties of the Lagrange interpolation polynomials on the real axis. Let \(C_{01}^{(r)}, r\geq 0, C_{01}^{(0)}= C_{01}\) be the space of functions \(f(x)\in C^{r}\) which has the form \(f(x)=2if_{1}(x)/(x+i)\), where \(f_{1}(x)\in C^{r}\). The Lagrange interpolation polynomial has the form \((L_{n}f)(x)= \sum\limits_{k=-n}^{n}a_{k}\omega_{k}(x)\), \(a_{k}= (2n+1)^{-1}\sum\limits_{j=-n}^{n}f(x_{j})\exp\{-{2\pi i\over 2n+1}jk\}\), where \(\omega_{k}(x)=[(x-i)/(x+i)]^{k}\), \(x_{j}=-{\text ctg}{\pi\over 2n+1}j, j=\overline{-n,n}\). One of the proved theorem is the following. If \(f(x)\in C_{01}\), then \((L_{n}f)(x)\in C_{01}\) and has the form \((L_{n}f)(x)=-\sum\limits_{k=-n}^{n}\gamma_{k}\psi_{k}(x)\), where \(\psi_{k}(x)=2i\omega_{k}(x)/(x+i), k=0,\pm1,\ldots\), \(\gamma_{k}= a_{n}+a_{n-1}+\ldots+a_{k+1}\). The authors study approximate properties of the Lagrange interpolation polynomials in the spaces \(C_{01}, L_{2}\) and prove the convergence of polynomial quadrature formulas of approximate calculation of singular Cauchy integral.