On an Asymptotic Equality for Reproducing Kernels and Sums of Squares of Orthonormal Polynomials

A. Ignjatovic () and D. S. Lubinsky ()
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A. Ignjatovic: University of New South Wales
D. S. Lubinsky: Georgia Institute of Technology

A chapter in Progress in Approximation Theory and Applicable Complex Analysis, 2017, pp 129-144 from Springer

Abstract: Abstract In a recent paper, the first author considered orthonormal polynomials p n $$\left \{p_{n}\right \}$$ associated with a symmetric measure with unbounded support and with recurrence relation x p n x = A n p n + 1 x + A n − 1 p n − 1 x , n ≥ 0 . $$\displaystyle{ xp_{n}\left (x\right ) = A_{n}p_{n+1}\left (x\right ) + A_{n-1}p_{n-1}\left (x\right ),\quad n \geq 0. }$$ Under appropriate restrictions on A n $$\left \{A_{n}\right \}$$ , the first author established the identity lim n → ∞ ∑ k = 0 n p k 2 x ∑ k = 0 n A k − 1 = lim n → ∞ p 2 n 2 x + p 2 n + 1 2 x A 2 n − 1 + A 2 n + 1 − 1 , $$\displaystyle{ \lim _{n\rightarrow \infty }\frac{\sum _{k=0}^{n}p_{k}^{2}\left (x\right )} {\sum _{k=0}^{n}A_{k}^{-1}} =\lim _{n\rightarrow \infty }\frac{p_{2n}^{2}\left (x\right ) + p_{2n+1}^{2}\left (x\right )} {A_{2n}^{-1} + A_{2n+1}^{-1}}, }$$ uniformly for x in compact subsets of the real line. Here, we establish and evaluate this limit for a class of even exponential weights, and also investigate analogues for weights on a finite interval, and for some non-even weights.

Keywords: Orthogonal polynomials; Christoffel functions; Recurrence coefficients; 42C05 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/978-3-319-49242-1_7

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