A Favard type theorem for orthogonal polynomials on the unit circle from a three term recurrence formula

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,Turn MathJaxoffwith 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 mu 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 muand 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 mu is such that M0 is the size of its mass at z=1. An example is also provided to completely illustrate the results obtained.

szego polynomials; kernel polynomials; para-orthogonal polynomials; chain sequences; continued fractions

A Favard type theorem for orthogonal polynomials on the unit circle from a three term recurrence formula Articles