The representation of functions in terms of their divided differences at Chebyshev nodes and roots of unity

Abstract

For the infinite triangular arrays of points whose rows consist of (i) the nth roots of unity, (ii) the extrema of Chebyshev polynomials Tn(x) on [−1, 1], and (iii) the zeros of Tn(x), we consider the corresponding sequences of divided difference functionals (In)1∞ in the successive rows of these arrays. We investigate the totality of such functionals as well as the convergence of the generalized Taylor series ∑1∞(In) Pn-1(z) a function f, where the Pk are basic polynomials satisfying Ij+1 Pk= δjk. Explicit formulae are given for the basic polynomials involving the Mobius function (of number theory), and examples of non-trivial functions f for which Inf = 0, n = 1, 2, …, are constructed. © 1990 Oxford University Press.