On second kind polynomials associated with rational transformations of linear functionals

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Abstract

In this paper the following construction process of orthogonal polynomials on the unit circle is considered: Let u be a regular and hermitian linear functional and let {Φn} be the corresponding orthogonal polynomials sequence. We define a new linear functional L by means the following relation with u:λ (z - β) L = (z - α) u, α, β, λ ∈ C, λ ≠ 0 . In this situation we obtain conditions for the regularity of L, as well as the corresponding orthogonal polynomials sequence. Also, we give one explicit representation for the orthogonal polynomials sequence of the second kind associated to L. For the particular case when α = β, L becomes in the well-known modification of u by addition of a Dirac mass. This case will be studied with special attention. © 2009.

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Suárez, C. (2009). On second kind polynomials associated with rational transformations of linear functionals. Journal of Mathematical Analysis and Applications, 358(1), 148–158. https://doi.org/10.1016/j.jmaa.2009.04.058

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