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§18.31 Bernstein–Szegő Polynomials

Let ρ(x) be a polynomial of degree and positive when -1x1. The Bernstein–Szegő polynomials {pn(x)}, n=0,1,, are orthogonal on (-1,1) with respect to three types of weight function: (1-x2)-12(ρ(x))-1, (1-x2)12(ρ(x))-1, (1-x)12(1+x)-12(ρ(x))-1. In consequence, pn(cosθ) can be given explicitly in terms of ρ(cosθ) and sines and cosines, provided that <2n in the first case, <2n+2 in the second case, and <2n+1 in the third case. See Szegő (1975, §2.6).