On certain polynomial and other approximations to analytic functions

Abstract

1. The Lagrange interpolation polynomials. This paper deals largely with certain polynomial approximations to analytic functions of a complex variable, that are somewhat analogous to the Lagrange interpolation polynomials. The latter, it will be recalled, are defined as follows: Given an analytic function f(z) of the complex variable z, and a set of n points z = a1, a2, , a,n, the corresponding Lagrange interpolation polynomial is the polynomial of (n 1)th degree at most, which, in case no two of the a, are equal to each other, agrees withf(z) at the points z = a,, z = a2, . z =a,, while if some of the ai are equal to each other, it is the limit of the Lagrange interpolation polynomial corresponding to n points a! that are all distinct and are allowed to approach the points ai respectively. In the latter case, if (say) a1 occurs just ni times in the sequence a1, a2, , an, the corresponding Lagrange interpolation polynomial will have contact of at least order ni 1 withf (z) at z = a,, that is, its derivatives of order 0, 1, , n1 + will be equal to the corresponding derivatives of f(z) at that point. These polynomials are among the most familiar approximations to functions of a real or complex variable, and general theorems are known which prove their convergence to f(z) as n becomes infinite, for properly restricted points a1, a2, * i , an, .? The most familiar instance of these polynomials is undoubtedly the case when all ai have a common value a, since the Lagrange polynomials corresponding to the first n terms of the sequence a,, a2, , an, now reduce merely to the first n terms of the Taylor expansion of f(z) about the point z = a; the discussion of the convergence of the polynomials to f(z) belongs to the elements of the theory of functions of a complex variable. One other case where the convergence problem may be