Zeros of orthogonal polynomials

V. M. Badkov

Trudy Instituta matematiki i mehaniki, Function theory, Tome 11 (2005) no. 2, pp. 30-46

Résumé

Let $\{T_{\sigma,n}(\tau)\}_{n=0}^\infty$ be an orthonormal on $[0,2\pi]$, with respect to some measure $d\sigma(\tau)$, system of trigonometric polynomials obtained from the sequence $1,\sin\tau,\cos\tau,\sin2\tau,\cos2\tau,\dots$ by Schmidt's orthogonalization method. A formula is established for the increment, at a point of the unit circle, of the argument of an algebraic polynomial orthogonal on it with respect to measure $d\sigma(\tau)$. Using this formula, for $n>0$, it is proved that zeros of the polynomial $T_{\sigma,n}(\tau)$ are real and simple and that zeros of the linear combinations $aT_{\sigma,2n-1}(\tau)+bT_{\sigma,2n}(\tau)$ and $-bT_{\sigma,2n-1}(\tau)+aT_{\sigma,2n}(\tau)$ alternate if $a^2+b^2>0$. For a wide class of weights with singularities whose orders are defined by finite products of real powers of concave moduli of continuity, it is proved that there exist positive constants $C_1$ and $C_2$, depending only on the weight, such that the distance between neighboring zeros of an orthogonal (with this weight) trigonometric polynomial of order $n$ lies between $C_1n^{-1}$ and $C_2n^{-1}$. In the form of corollaries, we deduce both known and new results on zeros of polynomials orthogonal with respect to a measure on a segment (possibly infinite).