[Paper Review] CMV biorthogonal Laurent polynomials: Christoffel formulas for Christoffel and Geronimus perturbations

Gerardo Ariznabarreta, Manuel Mañas+1 more|arXiv (Cornell University)|Oct 6, 2016

Mathematical functions and polynomials42 references3 citations

TL;DR

This paper develops Christoffel-type formulas for CMV biorthogonal Laurent polynomials under two types of perturbations—Christoffel (multiplication by a Laurent polynomial) and Geronimus (division by a Laurent polynomial with added masses)—providing explicit connection formulas for the perturbed polynomials, norms, Christoffel–Darboux kernels, and second kind functions. For prepared Laurent polynomials, the formulas yield quasideterminantal expressions involving only 2n unperturbed quantities, with special emphasis on the unit circle case where dual representations exist using either the polynomial family or mixed kernels.

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 $L(z)=L_nz^n+\cdots+L_{-n}z^{-n}$, $L_nL_{-n} eq 0$, 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.

Motivation & Objective

• To extend Christoffel and Geronimus transformations to the setting of CMV biorthogonal Laurent polynomials on the complex plane.
• To derive explicit connection formulas for perturbed biorthogonal Laurent polynomials, their norms, and Christoffel–Darboux kernels under both types of perturbations.
• To analyze the special case of the unit circle, where dual representations of perturbed quantities emerge via the nonperturbed family or mixed kernels.
• To provide quasideterminantal formulas expressing perturbed polynomials in terms of only 2n unperturbed objects for prepared Laurent polynomials.
• To demonstrate the formulas on examples including the real line, circle, Cassini oval, and cardioid, with detailed analysis on the unit circle.

Proposed method

• The study employs quasidefinite sesquilinear forms on Laurent polynomials, generalizing orthogonalities to include Sobolev and discrete types.
• Two Christoffel-type transformations are defined: multiplication by a Laurent polynomial and multiplication by its complex conjugate.
• Two Geronimus-type transformations are introduced: division by a Laurent polynomial and division by its complex conjugate, each accompanied by added Dirac masses at the zeros of the perturbing polynomial.
• The connection formulas are derived using spectral jets, second kind functions, and mixed Christoffel–Darboux kernels, leveraging matrix inversion and inner product identities.
• For prepared Laurent polynomials (symmetric degree support), the formulas reduce to quasideterminantal expressions involving determinants of matrices built from unperturbed polynomials and kernels.
• The unit circle case is analyzed in depth, revealing dual expressions for perturbed quantities—one using only the original biorthogonal family, the other using Christoffel–Darboux kernels and their mixed variants.

Experimental results

Research questions

• RQ1 How can Christoffel transformations be generalized to CMV biorthogonal Laurent polynomials under multiplication by Laurent polynomials and their conjugates?
• RQ2 What are the connection formulas for Geronimus transformations involving division by Laurent polynomials and their conjugates, including mass point additions?
• RQ3 How do the norms, Christoffel–Darboux kernels, and second kind functions transform under these perturbations?
• RQ4 What quasideterminantal formulas emerge for prepared Laurent polynomials, and how do they relate to the degree of the perturbing polynomial?
• RQ5 What unique dual representations arise in the unit circle case, and how do they reflect the role of reciprocal polynomials?

Key findings

• For prepared Laurent polynomials of degree n, the perturbed biorthogonal Laurent polynomial of any degree can be expressed as a linear combination of exactly 2n unperturbed polynomials, their second kind functions, or Christoffel–Darboux kernels.
• The connection formulas for all four transformation types (Christoffel and Geronimus, each with two variants) are derived in closed form using matrix inversion and inner product structures.
• In the unit circle case, the perturbed quantities admit two distinct representations: one using only the unperturbed biorthogonal family, and another using mixed Christoffel–Darboux kernels.
• The formulas are quasideterminantal, meaning they express the perturbed quantities as ratios of determinants built from unperturbed objects, preserving a constant number of terms independent of the degree.
• The second kind functions and mixed Christoffel–Darboux kernels are explicitly connected to the perturbed system, enabling full reconstruction of spectral data.
• Examples on the real line, circle, Cassini oval, and cardioid confirm the general framework, with the unit circle case showing exceptional symmetry due to the role of reciprocal polynomials.