Citation:
Journal of Approximation Theory, 2014, v. 184, pp. 146-162

ISSN:
0021-9045

DOI:
10.1016/j.jat.2014.05.007

Sponsor:
The works of all four authors have been supported by grants from CAPES, CNPq and FAPESP of Brazil. K. Castillo has also received support from Dirección General de Investigación, Ministerio de Economía y Competitividad of Spain (Grant MTM2012–36732–C03–01) for his research

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 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.[+][-]