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

Abstract

The objective of this manuscript is to study directly the Favard type theorem associated with the three term recurrence formulaRn+1(Z) = [(1 + ic(n+i))z + (1 - ic(n+1))]R-n(z) - 4d(n+1)zR(n-1)(z), n >= 1,with R-0(z) = 1 and R-1(z) = (1 + ic(1))z + (1 - ic(1)), where {c(n)}(n=1)(infinity) is a real sequence and {d(n)}(n=1)(infinity) is a positive chain sequence. We establish that there exists a unique nontrivial probability measure mu on the unit circle for which {R-n(z) - 2(1 - m(n))Rn-1(Z)} gives the sequence of orthogonal polynomials. Here, {m(n)}(n=0)(infinity) is the minimal parameter sequence of the positive chain sequence {d(n)}(n=1)(infinity). The element d(1) of the chain sequence, which does not affect the polynomials R-n, has an influence in the derived probability measure mu and hence, in the associated orthogonal polynomials on the unit circle. To be precise, if {M-n}(n=0)(infinity) is the maximal parameter sequence of the chain sequence, then the measure mu is such that M-0 is the size of its mass at z = 1. An example is also provided to completely illustrate the results obtained.