Universal relations in asymptotic formulas for orthogonal polynomials

TL;DR

This paper uncovers universal relations between amplitude and phase in the asymptotic behavior of orthogonal polynomials associated with Jacobi operators, independent of specific coefficient assumptions.

Contribution

It introduces universal relations linking amplitude and phase factors in asymptotic formulas for orthogonal polynomials, using a novel approach involving diagonalizing operators.

Findings

Universal amplitude-phase relations established

Conditions for boundedness of diagonalizing operators derived

Asymptotic formulas for orthogonal polynomials generalized

Abstract

Orthogonal polynomials $P_{n}(\lambda)$ are oscillating functions of $n$ as $n\to\infty$ for $\lambda$ in the absolutely continuous spectrum of the corresponding Jacobi operator $J$. We show that, irrespective of any specific assumptions on coefficients of the operator $J$, amplitude and phase factors in asymptotic formulas for $P_{n}(\lambda)$ are linked by certain universal relations found in the paper. Our approach relies on a study of operators diagonalizing Jacobi operators. Diagonalizing operators are constructed in terms of orthogonal polynomials $P_{n}(\lambda)$. They act from the space $L^2 (\Bbb R)$ of functions into the space $\ell^2 ({\Bbb Z}_{+})$ of sequences. We consider such operators in a rather general setting and find necessary and sufficient conditions of their boundedness.