Simultaneous Interpolation and Approximation by Entire Functions of Exponential Type

Abstract

Given the values of a function and possibly the values of some of its derivatives, at certain points, a practical problem of numerical analysis is to use this information to construct other functions that approximate it. Simultaneous interpolation and approximation of continuous functions on a compact interval, by polynomials, has been extensively studied by Runge, Bernstein, Faber, Fejer, Turan and others. Given a continuous function f on [−1,1] what is more natural than to think that a sequence of polynomials {pn (f;・)} which duplicate the function at n+1 equally spaced points of the interval will converge uniformly to f as n→∞.