Universal relations in asymptotic formulas for orthogonal polynomials

2020/11/30 by D. R. Yafaev, Yafaev, D. R.
Mathematics · #Spectral Theory in Mathematical Physics #Mathematical functions and polynomials #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2011.14987

Abstract

Orthogonal polynomials P_n(λ) are oscillating functions of n as n→∞ for λ 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(λ) 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(λ). They act from the space L² (\Bbb R) of functions into the space ℓ² (\Bbb Z₊) of sequences. We consider such operators in a rather general setting and find necessary and sufficient conditions of their boundedness.

Related

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• Spectral analysis of Jacobi operators and asymptotic behavior of orthogonal polynomials
• Asymptotic behavior of orthogonal polynomials without the Carleman condition
• Asymptotic behavior of orthogonal polynomials. Singular critical case
• Semiclassical asymptotic behavior of orthogonal polynomials