Some optimality conditions for Chebyshev expansions

Abstract In this paper we investigate conditions under which approximation to continuous functions on [−1, 1] by series of Chebyshev polynomials is superior to approximation by other ultraspherical orthogonal expansions. In particular we derive conditions on the Chebyshev coefficients which guarantee that the Chebyshev expansion of the corresponding functions converges more rapidly than expansions in Legendre polynomials or Chebyshev polynomials of the second kind.