The Representation of Functions in Terms of Their Divided Differences at Chebyshev Nodes and Roots of Unity

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}f in the successive rows of these arrays. We investigate the totality of such functionals as well as the convergence of the generalized Taylor series Li°(A»/)^n-i() f° a function/, where the Pk are basic polynomials satisfying lH1Pk = Sjlc. Explicit formulae are given for the basic polynomials involving the Mobius function (of number theory), and examples of non-trivial functions/for which / „ / = 0, n = 1,2,..., are constructed. Introduction Let / b e a function defined on the distinct complex points zv ...,zk. Recall that if m1,...,mk are positive integers with Xf-i < = > there exists a unique pe&n-i {&k denotes the set of polynomials of degree at most k), p{z) = ao + a1z+... +an_1z ~, satisfying p«\Zj) =f\Zj), i = 0,\,...,mj-l;j = \,...,k. (0.1) (An assumption that / has the required derivatives at zj when mj > 1 is implicit in (0.1).) The leading coefficient of/?, that is, an_x, is called the divided difference off with respect to zv...,zn (where each zj appears m} times in this sequence). In a more familiar notation we have an_1 =f(z1} ...,zn), and it is clear that an_x is a symmetric function of zx, ...,zn. We shall also use the notation hf'= "n-v It is obvious that In is a linear functional which satisfies Inq = 0 if qe&mi m <n-\; and lnz ~ = 1. Note that if z, = ... = zn = 0, then Inf = f -(0)/(n1)! Let ft denote an infinite triangular array of complex numbers whose y'th row, j = 0,1,2,... , is 0 = (fi[,...,/%) and suppose that / i s a function defined on all the entries in ft. It is easy to see that, in view of the elementary properties of the divided difference functionals Inf=f(J}[ ~, ...,$l ~), which we have just mentioned, there exist unique basic polynomials, Pk€^k, k = 0,1,2,.. . , that are monic and satisfy rj+1Pk = Sjk, j,k = 0,\,2,.... (0.2) Received 14 April 1989. 1980 Mathematics Subject Classification (1985 Revision) 41A58. The research of the first author was conducted while visiting the University of South Florida. The research of the third author was supported, in part, by the National Science Foundation, under grant DMS-862-0098. J. London Math. Soc. (2) 42 (1990) 309-328 310 K. G. IVANOV, T. J. RIVLIN AND E. B. SAFF Thus {Pk(z), / fc+1}*_0 is a normalized biorthogonal system, and, given fi, each/defined on it has the biorthogonal expansion