Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1505.03993

Hermite-Padé approximants for a pair of Cauchy transforms with overlapping symmetric supports

Alexander I. Aptekarev·Walter Van Assche·Maxim L. Yattselev
View original source ↗Read source document ↗Export metadata ↓

Abstract

Hermite-Padé approximants of type II are vectors of rational functions with common denominator that interpolate a given vector of power series at infinity with maximal order. We are interested in the situation when the approximated vector is given by a pair of Cauchy transforms of smooth complex measures supported on the real line. The convergence properties of the approximants are rather well understood when the supports consist of two disjoint intervals (Angelesco systems) or two intervals that coincide under the condition that the ratio of the measures is a restriction of the Cauchy transform of a third measure (Nikishin systems). In this work we consider the case where the supports form two overlapping intervals (in a symmetric way) and the ratio of the measures extends to a holomorphic function in a region that depends on the size of the overlap. We derive Szegő-type formulae for the asymptotics of the approximants, identify the convergence and divergence domains (the divergence domains appear for Angelesco systems but are not present for Nikishin systems), and show the presence of overinterpolation (a feature peculiar for Nikishin systems but not for Angelesco systems). Our analysis is based on a Riemann-Hilbert problem for multiple orthogonal polynomials (the common denominator).

Explore related subjects

math.CAmath.CV

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Alexander I. Aptekarev, Walter Van Assche, Maxim L. Yattselev. 2015-05-15. Hermite-Padé approximants for a pair of Cauchy transforms with overlapping symmetric supports. https://doi.org/10.1002/cpa.21675

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Positive Bidiagonal Factorizations: Combinatorial Structure, Spectral Theory, and Matrix Continued Fractions

Prescribed positive bidiagonal factorizations of semi-infinite banded matrices reveal spectral, approximation, and integrable structures for arbitrary lower and upper bandwidths. Every cyclic permutation of the factors admits a normalized lower--upper positive factorization. Under suitable degree conditions, factor transfers induce matrix Christoffel transformations, linking the cyclic Darboux orbit to mixed-type Favard theory. Christoffel words determine minimum-height positive refactorizations and the cyclic products that reach them. The associated matrices have two block displacements, reducing to two diagonals for coprime bandwidths. Their finite nonzero spectra lie on a star, and planar networks yield positive radial Stieltjes moment sequences. In the compact radial case star-supported representing measures are characterized by conditions at the origin and negative fractional moments. For coprime bandwidths, a radial moment deformation gives determinant solutions of a sparse Lax hierarchy and synchronizes the cyclic Toda flows. Retaining the prescribed factor order yields matrix continued fractions with explicit Padé-type contact, denominator factorizations, backward evaluation, and error bounds. For bounded nonnegative factors, the convergents are monotone and converge to the least nonnegative solution in the admissible domain. Under the Favard identification, they approximate the mixed-type Weyl matrix. Unbounded factorizations retain the formal approximation; analytic convergence requires a closed realization and stability hypotheses, with an additional identification for measure-defined Weyl matrices. Piñeiro and Jacobi-like systems give applications, including global positive coefficientwise integrable solutions under the stated positivity and AT hypotheses.

math.CA↗
arXiv2026

Spectrality of Weighted Measures on Two Line Segments

We study the spectrality of measures with positive integrable densities supported on two line segments in $\mathbb R^d$. We prove that, when the two segments are non-overlapping, spectrality forces the density on each segment to be constant almost everywhere. When the two segments are overlapping, spectrality forces the total density to be constant almost everywhere on their union. We then study the resulting measures with positive constant densities according to the geometric of the segments. For two non-coplanar segments, every such measure admits a spectrum contained in a straight line. For two segments lying on distinct parallel lines, the measure is spectral if and only if the two densities are equal. For non-parallel coplanar segments, a suitable invertible linear transformation converts the measure into an unweighted arc-length measure. When these segments are viewed in their affine plane, every spectrum of a spectral measure is contained in a straight line. Finally, we give examples showing how the densities determine the directions of line spectra and construct explicit spectra for weighted measures.

math.CA↗
arXiv2026

Spherical harmonics, operators of multiplication by coordinates, and infinitesimal conformal transformations

Consider the space of $C^\infty$-functions on the two-dimensional sphere $S^2$ and its decomposition $\oplus\mathcal H_n$ into a direct sum of minimal rotation-invariant spaces. We consider elements of $\oplus\mathcal H_n$ as functions of two variables, a nonnegative integer variable $n$ and a complex variable $u$ (a restriction of such function to the set $n=k$ is a polynomial in $u$ of degree $\le 2k$). For operators of multiplication by $x_1$, $x_2$, $x_3$ in $C^\infty(S^2)$ we obtain the corresponding operators in $\oplus\mathcal H_n$, they are differential-difference operators in the variables $u$, $n$ (including second derivatives in $u$ and shifts $n\mapsto n\pm1$). We obtain the similar correspondence for operators of differentiation along conformal vector fields on $S^2$.

math.CA↗