Publication

Journal of the London Mathematical Society

Paper

# 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.