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(1,1) - Coherent pairs on the unit circle

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2013-01-01
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Hindawi
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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.
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Sobolev orthogonal polynomials, Delta-Coherent pairs, n-Coherent Pairs, Szego theory, Extensions, Respect
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Abstract and Applied Analysis. Volume 2013, Article ID 307974, 8 pages