Publication: (1,1) - Coherent pairs on the unit circle
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Publication date
2013-01-01
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Tutors
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Publisher
Hindawi
Abstract
A pair (U,V) of Hermitian regular linear functionals on the unit circle is said to be a (1, 1)-coherent pair if their corresponding
sequences of monic orthogonal polynomials {𝜙𝑛(𝑥)}𝑛≥0 and {𝜓𝑛(𝑥)}𝑛≥0 satisfy 𝜙[1]
𝑛 (𝑧) + 𝑎𝑛𝜙[1]
𝑛−1(𝑧) = 𝜓𝑛(𝑧) + 𝑏𝑛𝜓𝑛−1(𝑧), 𝑎𝑛 ≠ 0, 𝑛 ≥ 1,
where 𝜙[1]
𝑛 (𝑧) = 𝜙
𝑛+1(𝑧)/(𝑛 + 1). In this contribution, we consider the cases when U is the linear functional
associated with the
Lebesgue and Bernstein-Szeg˝o measures, respectively, and we obtain a classification of the situations where V is associated with
either a positive nontrivial measure or its rational spectral transformation.
Description
Keywords
Sobolev orthogonal polynomials, Delta-Coherent pairs, n-Coherent Pairs, Szego theory, Extensions, Respect
Bibliographic citation
Abstract and Applied Analysis. Volume 2013, Article ID 307974, 8 pages