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First and second kind paraorthogonal polynomials and their zeros

Wong, Manwah Lilian (2007) First and second kind paraorthogonal polynomials and their zeros. Journal of Approximation Theory, 146 (2). pp. 282-293. ISSN 0021-9045. doi:10.1016/j.jat.2006.12.007.

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Given a probability measure μ with infinite support on the unit circle ∂D = {z: |z| = 1}, we consider a sequence of paraorthogonal polynomials h_n(z,λ) vanishing at z = λ where λ ∈ ∂D is fixed. We prove that for any fixed z_0 ∉ supp(dμ) distinct from λ, we can find an explicit ρ > 0 independent of n such that either h_n or h_(n+1) (or both) has no zero inside the disk B(z_0,ρ), with the possible exception of λ. Then we introduce paraorthogonal polynomials of the second kind, denoted s_n(z,λ). We prove three results concerning s_n and h_n. First, we prove that zeros of s_n and h_n interlace. Second, for z_0 an isolated point in supp(dμ), we find an explicit radius ρ(over tilde) such that either s_n or s_(n+1) (or both) have no zeros inside B(z_0, ρ(over tilde)). Finally, we prove that for such z_0 we can find an explicit radius such that either h_n or h_(n+1) (or both) has at most one zero inside the ball B(z_0, ρ(over tilde)).

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Additional Information:© 2007 Elsevier B.V. Received 18 May 2006; revised 5 December 2006; accepted 16 December 2006. Communicated by Leonid Golinskii. Available online 12 January 2007. Supported by the Croucher Foundation Scholarship, Hong Kong. I would like to thank Professor Barry Simon for his suggesting this problem, as well as his time for many very helpful discussions and email communications. I would also like to thank Cherie Galvez for her editorial advice as well as her help with LaTeX.
Funding AgencyGrant Number
Croucher FoundationUNSPECIFIED
Subject Keywords:paraorthogonal polynomials; second kind paraorthogonal polynomials; interlacing zeros; zeros of consecutive paraorthogonal polynomials
Issue or Number:2
Record Number:CaltechAUTHORS:20091118-092313714
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:16740
Deposited By: Jason Perez
Deposited On:18 Nov 2009 18:57
Last Modified:08 Nov 2021 23:29

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