Some Special Polynomials

Abstract

Chebyshev polynomials are introduced and their central role in problems in the uniform norm on [−1, 1] is explored. Sequences of orthogonal functions are then examined in some generality, although our primary interest is in orthogonal polynomials (and rational functions). The third section of this chapter is concerned with orthogonal polynomials; it introduces the most classical of these. These polynomials satisfy many extremal properties, similar to those of the Chebyshev polynomials, but with respect to (weighted) L ₂ norms. The final section of the chapter deals with polynomials with positive coefficients in various bases.