Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4607162 | Journal of Approximation Theory | 2014 | 17 Pages |
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+(1âicn+1)]Rn(z)â4dn+1zRnâ1(z),nâ¥1, with R0(z)=1 and R1(z)=(1+ic1)z+(1âic1), where {cn}n=1â is a real sequence and {dn}n=1â is a positive chain sequence. We establish that there exists a unique nontrivial probability measure μ on the unit circle for which {Rn(z)â2(1âmn)Rnâ1(z)} gives the sequence of orthogonal polynomials. Here, {mn}n=0â is the minimal parameter sequence of the positive chain sequence {dn}n=1â. 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=0â 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.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
K. Castillo, M.S. Costa, A. Sri Ranga, D.O. Veronese,