The degree of approximation by polynomials on some disjoint intervals in the complex plane

  • Authors:

  • Maurice Hasson

  • Affiliations:

  • Program in Applied Mathematics, The University of Arizona, Tucson, AZ 85721-0089, USA

  • Venue:
  • Journal of Approximation Theory
  • Year:
  • 2007

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Abstract

Let f(z) be a continuous function defined on the compact set K@?C and let E"n(f)=E"n(f,K) be the degree of approximation to f, for the supremum norm on K, by polynomials of degree (at most) n. ThusE"n(f,K)=infP@?P"n@?f-P@?.Here P"n denotes the space of polynomials of degree at most n and @?.@? is the supremum norm on K. For a positive integer s and for 0~E"n(f,K)^1^n=b^s^2-a^s^2b^s^2+a^s^2s,thus recovering several classical results. The proof of this error estimate is then translated into an algorithm that finds the polynomial of near best approximation.