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

  • A. Ignjatovic
  • D. S. Lubinsky
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 117)

Abstract

In a recent paper, the first author considered orthonormal polynomials \(\left \{p_{n}\right \}\) associated with a symmetric measure with unbounded support and with recurrence relation
$$\displaystyle{ xp_{n}\left (x\right ) = A_{n}p_{n+1}\left (x\right ) + A_{n-1}p_{n-1}\left (x\right ),\quad n \geq 0. }$$
Under appropriate restrictions on \(\left \{A_{n}\right \}\), the first author established the identity
$$\displaystyle{ \lim _{n\rightarrow \infty }\frac{\sum _{k=0}^{n}p_{k}^{2}\left (x\right )} {\sum _{k=0}^{n}A_{k}^{-1}} =\lim _{n\rightarrow \infty }\frac{p_{2n}^{2}\left (x\right ) + p_{2n+1}^{2}\left (x\right )} {A_{2n}^{-1} + A_{2n+1}^{-1}}, }$$
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 

2000 Mathematics Subject Classification

42C05 

Notes

Acknowledgements

The research of second author supported by NSF grant DMS136208.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.School of Computer Science and EngineeringUniversity of New South WalesSydneyAustralia
  2. 2.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

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