Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2611.00457

Bilateral Christoffel-Geronimus-Uvarov formulas for mixed-type orthogonality on algebraic supports

Manuel Mañas
View original source ↗Read source document ↗Export metadata ↓

Abstract

Christoffel-type determinant formulas construct both normalized polynomial families and their reproducing kernel for mixed-type orthogonality in several variables after independent rational matrix transformations on the two arguments. The pairing may have a rectangular matrix of functionals, with separate degree bounds on each component, and may be supported on an algebraic variety. The reduced matrices are calculated from the original biorthogonal polynomials, rational moments, polynomial division and the masses left free by division; their nonvanishing determinants characterize existence and uniqueness. For measure pairings the rational data are expressed by Cauchy and multivariate Fantappie transforms. Exact dimension counts incorporate the four matrix factors and the relations of the support. For normal degree-ordered original families and fixed factors, when the component degrees on each side differ by a bounded amount and the normalization dimension grows at most like one degree layer, the resulting orders are bounded on curves and grow at most linearly on surfaces. For pairings allowing scalar factors to move between arguments, called balanced pairings, an exact obstruction determines whether a bilateral mass decomposes into unilateral masses when the denominators act injectively. Residue duality provides coordinates for division and mass quotients on integral curves, including singular points, and on the finite surface quotients specified here. Explicit singular-curve families retain nondegenerate residue pairings as denominator zeros coalesce. Positive congruences admit a further reduction by orthogonal projection, while balanced pairings give simultaneous intertwining identities for all coordinate multiplication matrices.

Explore related subjects

math.CAmath-phmath.AGmath.MPmath.SP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Manuel Mañas. 2026-10-30. Bilateral Christoffel-Geronimus-Uvarov formulas for mixed-type orthogonality on algebraic supports. https://arxiv.org/abs/2611.00457

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

KEEP EXPLORING

Related papers

arXiv2026

Attainable forms of lower spectra

Let $d\in\mathbb{N}$ and $φ\colon(0,1)\to[0,d]$. We prove there exists a set $F\subset\mathbb{R}^d$ whose lower spectrum $\operatorname{dim}^θ_{\mathrm{L}} F$ satisfies $(1-θ)\operatorname{dim}^θ_{\mathrm{L}} F = φ(θ)$ for all $θ\in(0,1)$ if and only if for all $λ,θ\in(0,1)$, \begin{equation*} φ(θ) \leq φ(λθ) - θφ(λ) \leq (1-θ) d. \end{equation*} We also obtain a similar classification result for $\underline{\operatorname{dim}}^θ_{\mathrm{L}} F$. In contrast to the case for Assouad spectra, it is insufficient to consider homogeneous (or uniform) sets. Instead, we follow the approach introduced by Orgoványi--Rutar in arXiv:2510.07013 and proceed via a more general classification result for appropriate two-scale branching functions.

math.CA↗
arXiv2026

A $Tb$ type theorem for suppressed kernels

In this article, a non-homogeneous $Tb$ type theorem for arbitrary dimensional Calderón-Zygmund singular integral operators is proved. This is an extension of an analogous non-homogeneous $Tb$ theorem for the Cauchy transform, in the planar setting, due to Nazarov, Treil and Volberg. The novelties of the present work are the change of dimension and the fact that the operators to which the theorem applies are not necessarily antisymmetric. The techniques used in the proof include, among others, suppressed kernels, decompositions in $L^2(μ)$, where $μ$ is a Radon measure in $\mathbb{R}^d$, and a probabilistic argument resulting from taking averages of the operators involved.

math.CA↗
arXiv2026

Sparse domination modulo polynomials

We develop a sparse domination theory that captures both local polynomial approximation and cancellation of the input below the scale of each sparse cube. Its coefficients are percentiles of localized smooth maximal functions of the input minus its local polynomial approximation, combining features of mean-oscillation sparse bounds and of the cancellative sparse bounds of Conde-Alonso, Lorist and Rey. Using wavelet representations, we characterize the Calderón-Zygmund operators whose derivatives admit these bounds by explicit polynomial testing conditions. In the homogeneous setting, the same conditions characterize weighted Hardy-Sobolev inequalities with linear dependence on the $A_\infty$ characteristic. This quantitative dependence detects the transition from BMO to $L^\infty$ testing conditions that Lerner found for pointwise bounds by Riesz potentials, a distinction that qualitative weighted estimates cannot recover. In the inhomogeneous setting, we prove corresponding weighted inequalities and show that they are strictly weaker than pointwise sparse domination from the second derivative order onward. Consequences include Hardy-Sobolev estimates below the Banach range, sharp weighted Sobolev inequalities, and pointwise bounds by Riesz potentials.

math.CA↗