# §18.31 Bernstein–Szegö Polynomials

Let $\rho(x)$ be a polynomial of degree $\ell$ and positive when $-1\leq x\leq 1$. The Bernstein–Szegö polynomials $\{p_{n}(x)\}$, $n=0,1,\dots$, are orthogonal on $(-1,1)$ with respect to three types of weight function: $(1-x^{2})^{-\frac{1}{2}}(\rho(x))^{-1}$, $(1-x^{2})^{\frac{1}{2}}(\rho(x))^{-1}$, $(1-x)^{\frac{1}{2}}(1+x)^{-\frac{1}{2}}(\rho(x))^{-1}$. In consequence, $p_{n}(\mathop{\cos\/}\nolimits\theta)$ can be given explicitly in terms of $\rho(\mathop{\cos\/}\nolimits\theta)$ and sines and cosines, provided that $\ell<2n$ in the first case, $\ell<2n+2$ in the second case, and $\ell<2n+1$ in the third case. See Szegö (1975, §2.6).