Asymptotic behaviour of some families of orthonormal polynomials and an associated Hilbert space

TL;DR

This paper characterizes the asymptotic behavior of certain symmetric orthonormal polynomial families, explores their implications in a Hilbert space, and proposes a conjecture for non-symmetric cases supported by numerical evidence.

Contribution

It provides a detailed analysis of asymptotic properties of specific orthonormal polynomial families and their impact on the orthogonality of exponential functions in associated Hilbert spaces.

Findings

Symmetric orthonormal polynomials with specific recursion coefficients exhibit particular asymptotic behaviors.

In the related Hilbert space, exponential functions of different frequencies are mutually orthogonal.

Numerical tests support a conjecture about non-symmetric orthonormal polynomial families.

Abstract

We characterise asymptotic behaviour of families of symmetric orthonormal polynomials whose recursion coefficients satisfy certain conditions, satisfied for example by the (normalised) Hermite polynomials. More generally, these conditions are satisfied by the recursion coefficients of the form $c(n+1)^p$ for $0<p<1$ and $c>0$, as well as by recursion coefficients which correspond to polynomials orthonormal with respect to the exponential weight $W(x)=\exp(-|x|^\beta)$ for $\beta>1$. We use these results to show that, in a Hilbert space defined in a natural way by such a family of orthonormal polynomials, every two complex exponentials $e_{\omega}(t)={e}^{i \omega t}$ and $e_{\sigma}(t)={e}^{i \sigma t}$ of distinct frequencies $\omega,\sigma$ are mutually orthogonal. We finally formulate a surprising conjecture for the corresponding families of non-symmetric orthonormal polynomials; extensive numerical tests indicate that such a conjecture appears to be true.