Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.24100

Time-and-band limiting for symmetric classical Dunkl orthogonal polynomials

M. M. Castro·F. A. Grünbaum
View original source ↗Read source document ↗Export metadata ↓

Abstract

The problem of ''time and band limiting'' has been previously considered for expansions in orthonormal polynomials that are common eigenfunctions of some second order differential operator. In this paper we consider the situation of orthonormal polynomials that are eigenfunctions of some second order differential-difference operator as introduced by Ch. Dunkl. These polynomials appear already in the work of G. Szegő and at the same time play a role in supersymmetric quantum mechanics.

Explore related subjects

math.CAcs.NAmath.NAmath.SP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

M. M. Castro, F. A. Grünbaum. 2026-10-16. Time-and-band limiting for symmetric classical Dunkl orthogonal polynomials. https://arxiv.org/abs/2610.24100

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

KEEP EXPLORING

Related papers

arXiv2026

Positive Cubature Compression and Optimal Hyperinterpolation Stability on a Conic Surface

We study the computational cost and stability of positive cubature and hyperinterpolation on a truncated conic surface with a Jacobi-type weight. A classical quadratic disk-to-cone map is formulated as a measure-preserving $\mathbb Z_2$ quotient. It identifies degree-$m$ conic traces with even disk polynomials of degree at most $2m$ and gives a two-way correspondence between positive degree-$2n$ cone cubature and centrally symmetric positive degree-$(4n+1)$ disk cubature. Consequently, Möller's lower bound and the node excess above it transfer exactly, producing compressed non-product cone rules from near-minimal disk formulas. The quotient also transfers reproducing kernels and hyperinterpolation operators. Every positive degree-$2n$ cone rule yields an exact $L^2$ sampling isometry on $Π_n(V)$, so the weighted sampling matrix has condition number one, independently of the number and geometry of its nodes. The associated Lebesgue constants satisfy the rule-independent bound $Λ_{n,γ}^V\ge c_γn^{γ+1}$ for every $γ>-1$. For nonnegative half-integer $γ$, the matching upper bound holds, giving the sharp universal order $Λ_{n,γ}^V\asymp n^{γ+1}$; in particular, $Λ_n^V\asymp n$ when $γ=0$. Thus node compression preserves exact $L^2$ conditioning and, in this parameter range, optimal-order $L^\infty$ stability. Low-degree near-minimal rules and higher-degree optimized disk schemes illustrate the reduction in sampling cost.

math.CA↗
arXiv2026

A Sharp Lower Bound for Integral Menger Curvature

We prove that round circles $C_L$ are the unique minimisers of integral Menger curvature among embedded closed $C^1$ curves $γ$ of prescribed length $L$ for every $p\geq 2$, i.e. $M_p(C_L) \leq M_p(γ)$, with equality if and only if $γ$ parametrises a round circle. The proof combines a three-point symmetrisation, an analysis of diagonal truncations, and the periodic Wirtinger inequality. For $p>3$, known self-avoidance and regularity results extend the minimisation result to closed rectifiable curves without assuming embeddedness or smoothness, provided the length of the curve image equals $L$. In particular, $M_p$ detects the unknot and is basic for $p>3$, thereby answering a question posed repeatedly in the literature.

math.CA↗
arXiv2026

The $(n-2)$-linear restriction estimate in $\mathbb R^n$

We present a proof for the $(n-2)$-linear extension estimate for curved surfaces in $\mathbb{R}^n$, $n\ge5$, at the local product exponent $2(n-1)/(n-2)^2$, with an arbitrarily small power loss. The hypotheses are transversality of the normals and nondegeneracy of the compressed shape operators on their common two-dimensional orthogonal complement. The argument combines polynomial partitioning, a descent through algebraic varieties, localized multilinear restriction, and positive packet-energy estimates.

math.CA↗