Sharp Constants of Approximation Theory. IV. Asymptotic Relations in General Settings

Michael I. Ganzburg

Home/Archive/arXiv

arXiv · 2002.10512

Sharp Constants of Approximation Theory. IV. Asymptotic Relations in General Settings

Michael I. Ganzburg

View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we first introduce the unified definition of the sharp constant that includes constants in three major problems of approximation theory, such as, inequalities for approximating elements, approximation of individual elements, and approximation on classes of elements. Second, we find sufficient conditions that imply limit relations between various sharp constants of approximation theory in general settings. Third, a number of examples from various areas of approximation theory illustrates the general approach.

Explore related subjects

math.CA

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Michael I. Ganzburg. 2022-12-23. Sharp Constants of Approximation Theory. IV. Asymptotic Relations in General Settings. https://arxiv.org/abs/2002.10512

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

KEEP EXPLORING

Related papers

arXiv2026

Full-radius dimension-free maximal inequalities for discrete Euclidean balls

For every $1<p\le\infty$, we prove dimension-free maximal inequalities over all radii for normalized averages over Euclidean balls in $\mathbb Z^d$. In particular, this settles the $\ell^2$ question attributed to Stein. The proof uses a two-saddle expansion at integer squared radii to compare ball multipliers with normalized discrete Gaussians at the zero and parity frequencies. First-order estimates give the full-radius $\ell^2$ bound. For $1<p<2$, we combine higher-order residual estimates with dimension-uniform $\ell^1$ bounds for the residuals and levelwise interpolation to obtain the full range.

math.CA↗

arXiv2026

Polynomial corners in finite fields beyond the distinct-degree case

We prove a quantitative polynomial Roth theorem for corners in $\mathbb F_p^2$ for arbitrary pairs of linearly independent polynomials. More precisely, given a positive integer $d$, there are constants $p_0$ and $C$ (depending only on $p$) so that for every $ p>p_0$, if polynomials $ϕ_1,ϕ_2\in \mathbb \mathbb{F}_p [y]$ are of degree $\leq d$ vanishing at $0$ and are not linearly dependent, then every $A\subset\mathbb F_p^2$ with $ |A|\ge C p^{2-1/14} $ contains a nontrivial corner $$ (x_1,x_2),\qquad (x_1+ϕ_1(y),x_2),\qquad (x_1,x_2+ϕ_2(y)) $$ for some $y\in\mathbb F_p^\times$. This improves the estimate $p^{2-1/16}$ of Han--Lacey--Yang and removes the distinct-degree restriction from their quantitative theorem. The main obstruction is the equal-degree resonant case, where the Jacobian argument of Han--Lacey--Yang degenerates. We adjoin the frequency-independent part of the phase to form an augmented map $\widetilde F:W\to\mathbb A^3$ from the correlation threefold. We prove that this map is generically finite on every top-dimensional geometric component and has no two-dimensional fibre. Using the associated Artin--Schreier sheaf and Katz--Laumon estimates for Fourier transform of perverse sheaves, we obtain square-root cancellation outside an algebraic exceptional set of dimension at most one and uniformly bounded degree. A separate curve-sum argument gives uniform control on the exceptional set. An $\ell^2$ matrix estimate adapted to such sets completes the resonant case.

math.CA↗

arXiv2026

Symbolic Integration of Differential Forms: From Abel to Zeilberger

This paper focuses on symbolic integration of differential forms, with a particular emphasis on historical and modern developments, from Abel's addition theorems for Abelian integrals to Zeilberger's creative telescoping for parameterized integrals. It explores closed rational $p$-forms and provides algorithmic approaches for their integration, extending classical results like Hermite reduction and Liouville's theorem. The integration of closed differential forms with parameters is further examined through telescopers, offering a unified framework for handling both algebraic and transcendental cases.

math.CA↗