Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2611.01561

A log-concavity criterion for finite nonnegative linearization

Leslie P. Polzer
View original source ↗Read source document ↗Export metadata ↓

Abstract

In a 2005 survey, R. Szwarc asked for criteria, stated in the recurrence coefficients, for nonnegative linearization (property (P)) of finite orthogonal polynomial systems such as the Krawtchouk polynomials. For a finite three-term recurrence normalized to have row sums one, whose diagonal and downward coefficients have a constant nonnegative ratio, we prove that property (P) holds for every product, including products beyond the terminal degree, whenever the Newton coefficients of the profile $d_i=b_i+c_i$ are log-concave without internal zeros. The criterion contains the full positive Krawtchouk range and the dual $q$-Krawtchouk arrays, and through a quadratic decomposition of symmetric finite systems it covers Szwarc's quadratic system $Q_n$ in its sharp range $0<p_0\leq1/2$. As a limit to any such criterion, property (P) fails at coefficient distance $O((1-r^2)^N)$ from the Krawtchouk array of size $N$ with fixed ratio $c_i/d_i=r<1$. All theorems, corollaries and propositions are formalized in Lean 4.

Explore related subjects

math.CAmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Leslie P. Polzer. 2026-11-01. A log-concavity criterion for finite nonnegative linearization. https://arxiv.org/abs/2611.01561

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↗