Contracted zero distributions of extremal polynomials associated with slowly decaying weights
This paper is concerned with the asymptotic behaviour of the zeros of polynomials that are extremal with respect to slowly decaying weights on \([0,\infty)\). A specific examples will indicate the nature of the work. Consider the log-normal weight function given by \(w^2(x)= \exp(-\gamma^2(\log x)^2)\), where \(\gamma> 0\). The corresponding extremal monic polynomial \(S_n(x, \gamma)\) of degree \(n\) satisfies the orthogonality condition \(\int^\infty_0 x^kS_n(x, \gamma) w^2(x) dx= 0\) for \(k= 0,\dots, n-1\), and its zeros \(x_{1,n},\dots, x_{n,n}\) are real and simple. Let \(\delta(x)\) denote the Dirac measure at \(x\). Then \(n^{-1} \sum^n_{j=1} \delta(x^{1/n}_{j,n})\) is shown to tend to a weak\(^*\) limit \(\rho\) as \(n\to\infty\). Further, the measure \(\rho\) is supported by the interval \([1,\exp(\gamma^{- 2})]\) and has density there given by \(d\rho/dx= \gamma^2 x^{-1}\). The arguments involve logarithmic potential theory.
scientific article; zbMATH DE number 1525173