Abstract
Let {P n } be a sequence of orthogonal polynomials with respect to the measuredμ on the unit circle and letP n =P n +Σ =1l j λ nj P n−j forn≥l, whereλ n,j ∈ ℂ. It is shown that the sequence of linear combinations {P n },n≥2l, is orthogonal with respect to a positive measuredσ if and only ifdσ is a Bernstein-Szegö measure anddμ is the product of a unique trigonometric polynomial and the Bernstein-Szegö measuredσ. Furthermore for a given sequence ofP n 's an algorithm for the calculation of the λ n,j 's is provided.
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Communicated by S. Seatzu
Supported by Dirección General de Investigación Cientifica y Técnica (DGICYT) of Spain and Österreichischer Akademischer Austauschdienst of Austria with grant 4B/1995.
Also supported by the Austrian Fonds zur Förderung der wissenschaftlichen Forschung, project-number P9267-PHY.
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Marcellán, F., Peherstorfer, F. & Steinbauer, R. Orthogonality properties of linear combinations of orthogonal polynomials. Adv Comput Math 5, 281–295 (1996). https://doi.org/10.1007/BF02124748
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DOI: https://doi.org/10.1007/BF02124748