A property of Chebyshev polynomials

Abstract

THEOREM. rf P is a polynomial of degree n with n distinct zeros in [ 1, l] and 1 P(cos(k+z)/ = 1, k = 0, l,..., n, (1) then either P(x) = m(x) or P(x) = -T,(x), where T,(x) = cos(n arc cos x) is the Chebyshev polynomial of degree n. This theorem answers affirmatively a problem posed by C. Micchelli and T. Rivlin at the conference on “Linear Operators and Approximation” held in Oberwolfach in the summer of 1971, (see [l, p. 4981). For the proof, we will use a lemma due to W. W. Rogosinski [2]. Throughout, we assume that P is a polynomial of degree n with y1 distinct zeros in [--I, 11, satisfying (1).