Publication: A Favard type theorem for orthogonal polynomials on the unit circle from a three term recurrence formula
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2014-08
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Elsevier
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Abstract
The objective of this manuscript is to study directly the Favard type theorem associated with the three term recurrence formula
Rn+1(z)=[(1+icn+1)z+(1−icn+1)]Rn(z)−4dn+1zRn−1(z),n≥1,
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with R0(z)=1 and R1(z)=(1+ic1)z+(1−ic1) , where {cn}∞n=1 is a real sequence and {dn}∞n=1 is a positive chain sequence. We establish that there exists a unique nontrivial probability measure μ on the unit circle for which {Rn(z)−2(1−mn)Rn−1(z)} gives the sequence of orthogonal polynomials. Here, {mn}∞n=0 is the minimal parameter sequence of the positive chain sequence {dn}∞n=1 . The element d1 of the chain sequence, which does not affect the polynomials Rn , has an influence in the derived probability measure μ and hence, in the associated orthogonal polynomials on the unit circle. To be precise, if {Mn}∞n=0 is the maximal parameter sequence of the chain sequence, then the measure μ is such that M0 is the size of its mass at z=1 . An example is also provided to completely illustrate the results obtained.
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Szegő polynomials, Kernel polynomials, Para-orthogonal polynomials, Chain sequences, Continued fractions
Bibliographic citation
Journal of Approximation Theory, 2014, v. 184, pp. 146-162