CMV biorthogonal Laurent polynomials: Christoffel formulas for Christoffel and Geronimus perturbations

Abstract

Quasidefinite sesquilinear forms for Laurent polynomials in the complex plane and corresponding CMV biorthogonal Laurent polynomial families are studied. Bivariate linear functionals encompass large families of orthogonalities like Sobolev and discrete Sobolev types. Two possible Christoffel transformations of these linear functionals are discussed. Either the linear functionals are multiplied by a Laurent polynomial, or are multiplied by the complex conjugate of a Laurent polynomial. For the Geronimus transformation, the linear functional is perturbed in two possible manners as well, by a division by a Laurent polynomial or by a complex conjugate of a Laurent polynomial, in both cases the addition of appropriate masses (linear functionals supported on the zeros of the perturbing Laurent polynomial) is considered. The connection formulas for the CMV biorthogonal Laurent polynomials, its norms, and Christoffel-Darboux kernels, in all the four cases, are given. For the Geronimus transformation, the connection formulas for the second kind functions and mixed Christoffel-Darboux kernels are also given in the two possible cases. For prepared Laurent polynomials, i.e. of the form $L(z)=L_nz^n+\cdots+L_{-n}z^{-n}$, $L_nL_{-n}\neq 0$, these connection formulas lead to quasideterminantal (quotient of determinants) Christoffel formulas for all the four transformations, expressing an arbitrary degree perturbed biorthogonal Laurent polynomial in terms of $2n$ unperturbed biorthogonal Laurent polynomials, their second kind functions or Christoffel-Darboux kernels and its mixed versions. The unit circle case, given its exceptional properties, is discussed in more detail.