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 formula % \[ R_{n+1}(z) = \big[(1+ic_{n+1})z+(1-ic_{n+1})\big] R_{n}(z) - 4 d_{n+1} z R_{n-1}(z), \quad n \geq 1, \] % with $R_0(z)=1$ and $R_1(z)=(1+i{c}_1)z+(1-i{c}_1)$, where $\{c_n{\}}_{n=1}^{\infty }$ is a real sequence and $\{d_n{\}}_{n=1}^{\infty }$ is a positive chain sequence. We establish that there exists an unique nontrivial probability measure $\mu$ on the unit circle for which $\{R_n(z)-2(1-m_n)R_{n-1}(z)\}$ gives the sequence of orthogonal polynomials. Here, $\{m_n{\}}_{n=0}^{\infty }$ is the minimal parameter sequence of the positive chain sequence $\{d_n{\}}_{n=1}^{\infty }$. The element $d_1$ of the chain sequence, which does not effect 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 $\{M_n{\}}_{n=0}^{\infty }$ 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 illustrates the results obtained.

15 pages, 1 figure

References in corpus (2)

• Equality of the Spectral and Dynamical Definitions of Reflection
• Elliptic Hypergeometric Laurent Biorthogonal Polynomials with a Dense Point Spectrum on the Unit Circle

Cited by in corpus (8)

• Para-orthogonal polynomials on the unit circle satisfying three term recurrence formulas
• Extreme zeros in a sequence of para-orthogonal polynomials and bounds for the support of the measure
• Christoffel formula for kernel polynomials on the unit circle
• Orthogonal Polynomials Associated with Complementary Chain Sequences
• Biorthogonality and para-orthogonality of $R_I$ polynomials
• Spectral transformation associated with a perturbed $R_I$ type recurrence relation
• Orthogonal polynomials on the unit circle: Verblunsky coefficients with some restrictions imposed on a pair of related real sequences
• Complementary Romanovski-Routh polynomials: From orthogonal polynomials on the unit circle to Coulomb wave functions