A generalization of the Whittaker-Kotel'nikov-Shannon sampling theorem for continuous functions on a~closed interval

A. Yu. Trynin

Sbornik. Mathematics, Tome 200 (2009) no. 11, pp. 1633-1679

Résumé

Classes of functions in the space of continuous functions $f$ defined on the interval $[0,\pi]$ and vanishing at its end-points are described for which there is pointwise and approximate uniform convergence of Lagrange-type operators $$ S_\lambda(f,x)=\sum_{k=0}^n\frac{y(x,\lambda)}{y'(x_{k,\lambda}) (x-x_{k,\lambda})}f(x_{k,\lambda}). $$ These operators involve the solutions $y(x,\lambda)$ of the Cauchy problem for the equation $$ y''+(\lambda-q_\lambda(x))y=0 $$ where $q_\lambda\in V_{\rho_\lambda}[0,\pi]$ (here $V_{\rho_\lambda}[0,\pi]$ is the ball of radius $\rho_\lambda=o(\sqrt\lambda/\ln\lambda)$ in the space of functions of bounded variation vanishing at the origin, and $y(x_{k,\lambda})=0$). Several modifications of this operator are proposed, which allow an arbitrary continuous function on $[0,\pi]$ to be approximated uniformly. Bibliography: 40 titles.

Détail

Keywords: sampling theorem, sinc approximation.
Mots-clés : interpolation, uniform convergence