Widom’s formula for the leading coefficient of a polynomial which is orthonormal with respect to a varying weight. (English. Russian original) Zbl 1281.42025

Komlov, A. V.; Suetin, S. P.

Russ. Math. Surv. 67, No. 1, 183-185 (2012); translation from Usp. Mat. Nauk 67, No. 1, 183-184 (2012).

Summary: We obtain a strong asymptotic formula for the leading coefficient \(\alpha_n(n)\) of a degree \(n\) polynomial \(q_n(z;n)\) orthonormal on a system of intervals on the real line with respect to a varying weight. The weight depends on \(n\) as \(e^{-2nQ(x)}\), where \(Q(x)\) is a polynomial and corresponds to the “hard-edge-case”. The formula in Theorem 1 is quite similar to Widom’s classical formula for a weight independent of \(n\). In some sense, Widom’s formulas are still true for a varying weight and are thus universal. As a consequence of the asymptotic formula we have that \(\alpha_n(n)e^{-nw_Q}\) oscillates as \(n\to\infty\) and, in a typical case, fills an interval (here \(w_Q\) is the equilibrium constant in the external field \(Q\)).

MSC:

42C05

Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis