The Analytic Character of the Birkhoff Interpolation Polynomials

Abstract

Let E be an m × (n + 1) regular interpolation matrix with elements ei, k = (E)i, k which are zero or one, with n + 1 ones. Then for each f ∈ Cn[a, b] and each set of knots X: a ≦ x1 < … < xm ≦ b, there is a unique interpolation polynomial P(f, E, X; t) of degree ≦ n which satisfies1A recent paper [1] discussed the continuity of P, as a function of x1, …,xm(with coalescences allowed). We would like to study in this note the analytic character of P as a function of real or complex knots X: x1, …, xm. This is easy for the Lagrange or the Hermite interpolation. In this case P is a polynomial in x1, …, xm if f is a polynomial, and an entire function in x1, …, xm if f is entire.