A proof of the Szeg\H{o} conjecture on Jacobi extrema

Abstract

For Jacobi polynomials with parameters greater than $-1/2$, and with the relative extrema enumerated from the endpoint $x=1$, the normalised modulus at the $k$th extremum of degree $n+1$ is proved to be strictly smaller than that at the $k$th extremum of degree $n$, for $1\leq k\leq n$. This proves the Szeg\H{o} conjecture, recorded in the 1975 fourth edition of his classic monograph Orthogonal Polynomials, and strengthens it by removing the ordering assumption on the parameters. The proof transforms the Jacobi equation to angular form and compares Pr\"ufer amplitudes at equal phase. Combined with a reduction of de Oliveira Filho and a separate quadratic-transformation argument for the boundary case, the result also settles a question concerning the Lov\'asz theta number of spherical distance graphs in every dimension at least four.