arXiv subjects

Marcus Vaktnäs

Publications and source records attributed to Marcus Vaktnäs.

7 recordsLinked to original sources

arXiv2026

Determinantal Formulas for Rational Perturbations of Multiple Orthogonality Measures

Given multiple orthogonal polynomials on the real line with respect to a system $\bmμ = (μ_1,\ldots,μ_r)$, we investigate multiple orthogonal polynomials associated with any rational perturbation of the form $$ \widetilde{\bmμ}=\Big(\frac{Φ_1}{Ψ_{1}} μ_1,\dots,\frac{Φ_r}{Ψ_r}μ_r\Big), $$ for any polynomials $Φ_1,\dots,Φ_r$ and $Ψ_1,\dots,Ψ_r$. We derive the analogues of Uvarov's determinantal formula for the multiple orthogonal polynomials of type I and type II for $\widetilde{\bmμ}$ and establish necessary and sufficient condition for normality of the indices. The result allows the polynomials $\{Φ_j,Ψ_j\}_{j=1}^r$ to be arbitrary and permits the addition of finitely many point masses to each of the measures $μ_j$. Moreover, the measures $μ_j$ may be taken as quasi-definite linear functionals, which is of interest even in the case $r=1$.

math.CA↗

arXiv2026

Zeros of Laurent multiple orthogonal polynomials on the unit circle

We investigate two distinct formulations of Laurent multiple orthogonal polynomials on the unit circle, introduced in arXiv:2410.12094 and arXiv:2601.04783 respectively. For the first formulation, we prove that all zeros lie strictly within the complex open unit disk for any Angelesco or AT system. For the second formulation, we establish normality of all indices of the form $(\bm{n};\bm{n})$, $(\bm{n}+\bm{e}_j;\bm{n})$, and $(\bm{n};\bm{n}+\bm{e}_j)$ for any Angelesco or AT system, thereby enabling the full application of the Szegő mapping and Geronimus relations from arXiv:2601.04783 in the multiple orthogonality setting.

math.CV↗

arXiv2026

Szegő Mapping and Hermite--Padé Polynomials for Multiple Orthogonality on the Unit Circle

We investigate generalized Laurent multiple orthogonal polynomials on the unit circle satisfying simultaneous orthogonality conditions with respect to $r$ probability measures or linear functionals on the unit circle. We show that these polynomials can be characterized as solutions of a general two-point Hermite--Padé approximation problem. We derive Szegő-type recurrence relations, establish compatibility conditions for the associated recurrence coefficients, and obtain Christoffel--Darboux formulas as well as Heine-type determinantal representations. Furthermore, by extending the Szegő mapping and the Geronimus relations, we relate these Laurent multiple orthogonal polynomials to multiple orthogonal polynomials on the real line, thereby making explicit the connection between multiple orthogonality on the unit circle and on the real line.

math.CA↗

arXiv2026

Angelesco and AT systems on the Unit Circle

We introduce the concept of Laurent multiple orthogonality on the unit circle and define Angelesco and AT systems in this setting. Using a generalized Andreief identity, we establish normality of all multi-indices for any such system, thereby ensuring existence and uniqueness of Laurent multiple orthogonal polynomials of type I and type II at every location. As an application, we demonstrate existence and uniqueness of the approximants for two natural two-point Hermite-Padé problems -- type I and type II -- arising in the simultaneous rational approximation of $r$ Carathéodory functions.

math.CA↗

arXiv2025

Christoffel Transform and Multiple Orthogonal Polynomials

We investigate multiple orthogonal polynomials associated with the system of measures obtained by applying a Christoffel transform to each of the orthogonality measures. We present an algorithm for computing the transformed recurrence coefficients and determinantal formulas for the transformed multiple orthogonal polynomials of type I and type II. We apply these results to show that zeros of multiple orthogonal polynomials of an Angelesco or an AT system interlace with the zeros of the polynomials corresponding to its one-step Christoffel transform. This allows us to prove a number of interlacing properties satisfied by the multiple orthogonality analogues of classical orthogonal polynomials. For the discrete polynomials, this also produces an estimate on the smallest distance between consecutive zeros. We also identify a connection between the Christoffel transform of orthogonal polynomials and multiple orthogonality systems containing a finitely supported measure. In consequence, the compatibility relations for the nearest neighbour recurrence coefficients provide a new algorithm for the computation of the Jacobi coefficients of the one-step or multi-step Christoffel transforms.

math.CA↗

arXiv2025

Zeros of Multiple Orthogonal Polynomials: Location and Interlacing

We prove a criterion on the possible locations of zeros of type I and type II multiple orthogonal polynomials in terms of normality of degree $1$ Christoffel transforms. We provide another criterion in terms of degree $2$ Christoffel transforms for establishing zero interlacing of the neighbouring multiple orthogonal polynomials of type I and type II. We apply these criteria to establish zero location and interlacing of type I multiple orthogonal polynomials for Nikishin systems. Additionally, we recover the known results on zero location and interlacing for type I multiple orthogonal polynomials for Angelesco systems, as well as for type II multiple orthogonal polynomials for Angelesco and AT systems. Finally, we demonstrate that normality of the higher order Christoffel transforms is naturally related to the zeros of the Wronskians of consecutive orthogonal polynomials.

math.CA↗

arXiv2024

Szegő Recurrence for Multiple Orthogonal Polynomials on the Unit Circle

We investigate polynomials that satisfy simultaneous orthogonality conditions with respect to several measures on the unit circle. We generalize the direct and inverse Szegő recurrence relations, identify the analogues of the Verblunsky coefficients, and prove the Christoffel$\unicode{x2013}$Darboux formula. These results stand directly in analogue with the nearest neighbour recurrence relations from the real line counterpart.

math.CA↗