Journal of Approximation Theory最新文献

英文中文

Optimization-aided construction of multivariate Chebyshev polynomials 优化辅助构建多元切比雪夫多项式

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-01-01 Epub Date: 2024-10-24 DOI: 10.1016/j.jat.2024.106116

M. Dressler , S. Foucart , M. Joldes , E. de Klerk , J.B. Lasserre , Y. Xu

This article is concerned with an extension of univariate Chebyshev polynomials of the first kind to the multivariate setting, where one chases best approximants to specific monomials by polynomials of lower degree relative to the uniform norm. Exploiting the Moment-SOS hierarchy, we devise a versatile semidefinite-programming-based procedure to compute such best approximants, as well as associated signatures. Applying this procedure in three variables leads to the values of best approximation errors for all monomials up to degree six on the euclidean ball, the simplex, and the cross-polytope. Furthermore, inspired by numerical experiments, we obtain explicit expressions for Chebyshev polynomials in two cases unresolved before, namely for the monomial

x_{1}^{2} x_{2}^{2} x_{3}

on the euclidean ball and for the monomial

x_{1}^{2} x_{2} x_{3}

on the simplex.

本文关注的是将第一类单变量切比雪夫多项式扩展到多变量环境，即通过相对于统一规范的低度多项式来追寻特定单项式的最佳近似值。利用 Moment-SOS 层次结构，我们设计了一种基于半定量编程的通用程序，用于计算此类最佳近似值以及相关签名。在三个变量中应用这一程序，就能得出欧几里得球、简单面和交叉多面体上六度以内所有单项式的最佳近似误差值。此外，在数值实验的启发下，我们还得到了切比雪夫多项式在两种情况下的明确表达式，即欧几里得球上的单项式 x12x22x3 和单纯形上的单项式 x12x2x3。

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引用次数: 0

Positive orthogonalizing weights on the unit circle for the generalized Bessel polynomials 广义贝塞尔多项式单位圆上的正交权重

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-01-01 Epub Date: 2024-10-22 DOI: 10.1016/j.jat.2024.106115

Sergey M. Zagorodnyuk

<div><div>In this paper we study the generalized Bessel polynomials <span><math><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></mrow></math></span> (in the notation of Krall and Frink). Let <span><math><mrow><mi>a</mi><mo>></mo><mn>1</mn></mrow></math></span>, <span><math><mrow><mi>b</mi><mo>∈</mo><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>/</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>/</mo><mn>3</mn><mo>)</mo></mrow><mo>∖</mo><mrow><mo>{</mo><mn>0</mn><mo>}</mo></mrow></mrow></math></span>. In this case we present the following positive continuous weights <span><math><mrow><mi>p</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow><mo>=</mo><mi>p</mi><mrow><mo>(</mo><mi>θ</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></mrow></math></span> on the unit circle for <span><math><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></mrow></math></span>: <span><math><mrow><mn>2</mn><mi>π</mi><mi>p</mi><mrow><mo>(</mo><mi>θ</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>1</mn><mo>+</mo><mn>2</mn><mrow><mo>(</mo><mi>a</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><msubsup><mrow><mo>∫</mo></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn></mrow></msubsup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mi>b</mi><mi>u</mi><mo>cos</mo><mi>θ</mi></mrow></msup><mo>cos</mo><mrow><mo>(</mo><mi>b</mi><mi>u</mi><mo>sin</mo><mi>θ</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>u</mi><mo>)</mo></mrow></mrow><mrow><mi>a</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>d</mi><mi>u</mi><mo>,</mo></mrow></math></span> where <span><math><mrow><mi>θ</mi><mo>∈</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>2</mn><mi>π</mi><mo>]</mo></mrow></mrow></math></span>. Namely, we have <span><math><mrow><msubsup><mrow><mo>∫</mo></mrow><mrow><mn>0</mn></mrow><mrow><mn>2</mn><mi>π</mi></mrow></msubsup><msub><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mi>θ</mi></mrow></msup><mo>,</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>m</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mi>θ</mi></mrow></msup><mo>,</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow><mi>p</mi><mrow><mo>(</mo><mi>θ</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow><mi>d</mi><mi>θ</mi><mo>=</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><msub><mrow><mi>δ</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>m</mi></mrow></msub><mo>,</mo><mspace></mspace><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>≠</mo><mn>0</mn><mo>,</mo><mspace></mspace><mi>n</mi><mo>,</mo><mi>m</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>.</mo></mrow></math></span> Notice that this orthogon

本文研究广义贝塞尔多项式 yn(x,a,b)（用 Krall 和 Frink 的符号表示）。设 a>1, b∈(-1/3,1/3)∖{0}。在这种情况下，我们在单位圆上为 yn(x,a,b) 提出以下正连续权值 p(θ)=p(θ,a,b) ：2πp(θ,a,b)=-1+2(a-1)∫01e-bucosθcos(businθ)(1-u)a-2du，其中θ∈[0,2π]。即，我们有∫02πyn(eiθ,a,b)ym(eiθ,a,b)p(θ,a,b)dθ=Cnδn,m,Cn≠0,n,m=0,1,2,....。注意，这个正交性与单位圆上正交多项式的通常正交性不同。给出了上述正交性的一些应用。

引用次数: 0

Translation-based completeness on compact intervals 紧凑区间上基于翻译的完备性

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-01-01 Epub Date: 2024-09-28 DOI: 10.1016/j.jat.2024.106104

Lukas Liehr

Given a compact interval

I \subseteq R

, and a function

f

that is a product of a nonzero polynomial with a Gaussian, it will be shown that the translates

{f (\cdot - λ) : λ \in Λ}

are complete in

C (I)

if and only if the series of reciprocals of

Λ

diverges. This extends a theorem in [R. A. Zalik, Trans. Amer. Math. Soc. 243, 299–308]. An additional characterization is obtained when

Λ

is an arithmetic progression, and the generator

f

constitutes a linear combination of translates of a function with sufficiently fast decay.

给定一个紧凑区间 I⊆R，以及一个非零多项式与高斯的乘积函数 f，将证明当且仅当 Λ 的倒数列发散时，平移 {f(⋅-λ):λ∈Λ} 在 C(I) 中是完全的。这扩展了[R. A. Zalik, Trans. Amer. Math. Soc. 243, 299-308] 中的定理。当Λ 是算术级数，且生成器 f 构成具有足够快衰减的函数平移的线性组合时，可以得到额外的特征。

引用次数: 0

Monotonicity of zeros of derivatives of Bessel functions 贝塞尔函数导数零点的单调性

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-01-01 Epub Date: 2024-09-26 DOI: 10.1016/j.jat.2024.106102

Dimitar K. Dimitrov, Yen Chi Lun

Recently Baricz et al., 2018 and Baricz and Singh 2018 gave two different proofs of the fact that the zeros of the

n

th derivative of the Bessel function of the first kind

J_{ν} (x)

are all real when

ν > n - 1

. We provide a third alternative proof. The authors of Baricz et al., 2018 conjectured that, for every

n \in N

, the positive zeros of

J_{ν}^{(n)} (x)

are increasing functions of the parameter

ν

, for

ν \in (n - 1, \infty)

. We provide two apparently distinct proofs of the conjecture.

最近，Baricz 等人 2018 年以及 Baricz 和 Singh 2018 年给出了两个不同的证明，证明了当 ν>n-1 时，第一类贝塞尔函数的 n 次导数 Jν(x) 的零点都是实数。我们提供了第三个替代证明。Baricz 等人，2018》的作者猜想，对于每 n∈N，Jν(n)(x) 的正零点是参数 ν 的递增函数，为 ν∈（n-1,∞）。我们提供了两个看似不同的猜想证明。

引用次数: 0

On reverse Markov–Nikol’skii inequalities for polynomials with restricted zeros 关于有限制零点的多项式的反向马尔可夫-尼克尔斯基不等式

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-01-01 Epub Date: 2024-08-30 DOI: 10.1016/j.jat.2024.106097

Mikhail A. Komarov

Let $Π_{n}$ be the class of algebraic polynomials $P$ of degree $n$ , all of whose zeros lie on the segment $[- 1, 1]$ . In 1995, S. P. Zhou has proved the following Turán type reverse Markov–Nikol’skii inequality: ${‖ P^{'} ‖}_{L_{p} [- 1, 1]} > c {(\sqrt{n})}^{1 - 1 / p + 1 / q} {‖ P ‖}_{L_{q} [- 1, 1]}$ , $P \in Π_{n}$ , where $0 < p \leq q \leq \infty$ , $1 - 1 / p + 1 / q \geq 0$ ( $c > 0$ is a constant independent of $P$ and $n$ ). We show that Zhou’s estimate remains true in the case $p = \infty$ , $q > 1$ . Some of related Turán type inequalities are also discussed.

设 Πn 是 n 阶代数多项式 P 的类，其所有零点都位于线段 [-1,1] 上。1995 年，S. P.周证明了下面的图兰型逆马尔科夫-尼克尔斯基不等式：P′‖Lp[-1,1]>c(n)1-1/p+1/q‖P‖Lq[-1,1]，P∈Πn，其中 0<p≤q≤∞，1-1/p+1/q≥0（c>0 是与 P 和 n 无关的常数）。我们还讨论了一些相关的图兰式不等式。

引用次数: 0

Lower bounds for piecewise polynomial approximations of oscillatory functions 振荡函数的片断多项式近似值下限

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-01-01 Epub Date: 2024-09-14 DOI: 10.1016/j.jat.2024.106100

Jeffrey Galkowski

We prove lower bounds on the error incurred when approximating any oscillating function using piecewise polynomial spaces. The estimates are explicit in the polynomial degree and have optimal dependence on the meshwidth and frequency when the polynomial degree is fixed. These lower bounds, for example, apply when approximating solutions to Helmholtz plane wave scattering problem.

我们证明了使用分次多项式空间逼近任何振荡函数时产生的误差下限。这些估计值在多项式阶数上是显式的，并且在多项式阶数固定时，与网格宽度和频率有最佳依赖关系。例如，这些下限适用于近似求解亥姆霍兹平面波散射问题。

引用次数: 0

Gaussian quadrature formulae are strongly asymptotically optimal for a class of infinitely differentiable functions 高斯正交公式是一类无限可微分函数的强渐近最优公式

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-01-01 Epub Date: 2024-10-24 DOI: 10.1016/j.jat.2024.106117

Guiqiao Xu

This paper investigates the optimal quadrature formulae of a class

F_{\infty}

of infinitely differentiable functions on

[- 1, 1]

. We obtain the strong equivalences of the optimal worst case errors of

F_{\infty}

for standard information and Hermite data. We proved that the Gaussian quadrature formulae are strongly asymptotically optimal.

本文研究了[-1,1]上一类无限微分函数 F∞ 的最优正交公式。我们得到了标准信息和赫米特数据下 F∞ 的最优最坏情况误差的强等价性。我们证明了高斯正交公式是强渐近最优的。

引用次数: 0

Function recovery on manifolds using scattered data 利用分散数据恢复流形上的函数

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-01-01 Epub Date: 2024-09-13 DOI: 10.1016/j.jat.2024.106098

David Krieg , Mathias Sonnleitner

We consider the task of recovering a Sobolev function on a connected compact Riemannian manifold $M$ when given a sample on a finite point set. We prove that the quality of the sample is given by the $L_{γ} (M)$ -average of the geodesic distance to the point set and determine the value of $γ \in (0, \infty]$ . This extends our findings on bounded convex domains [IMA J. Numer. Anal., 2024]. As a byproduct, we prove the optimal rate of convergence of the $n$ th minimal worst case error for $L_{q} (M)$ -approximation for all $1 \leq q \leq \infty$ .

Further, a limit theorem for moments of the average distance to a set consisting of i.i.d. uniform points is proven. This yields that a random sample is asymptotically as good as an optimal sample in precisely those cases with $γ < \infty$ . In particular, we obtain that cubature formulas with random nodes are asymptotically as good as optimal cubature formulas if the weights are chosen correctly. This closes a logarithmic gap left open by Ehler, Gräf and Oates [Stat. Comput., 29:1203-1214, 2019].

我们考虑了当给定有限点集上的样本时，在连通紧凑黎曼流形 M 上恢复 Sobolev 函数的任务。我们证明样本的质量是由到点集的大地距离的 Lγ(M)- 平均值给出的，并确定了 γ∈(0,∞] 的值。这扩展了我们在有界凸域上的发现[IMA J. Numer. Anal.］作为副产品，我们证明了 Lq(M)-approximation 在所有 1≤q≤∞ 条件下第 n 次最小最坏情况误差的最佳收敛速率。由此可以得出，正是在γ<∞的情况下，随机样本在渐近上与最优样本一样好。特别是，如果权重选择得当，我们可以得到带有随机节点的立体公式在渐近上与最优立体公式一样好。这弥补了埃勒、格拉夫和奥茨[Stat. Comput., 29:1203-1214, 2019]留下的对数差距。

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引用次数: 0

Singular examples of the Matrix Bochner Problem 矩阵波赫纳问题的奇异实例

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2024-12-01 Epub Date: 2024-07-26 DOI: 10.1016/j.jat.2024.106082

Ignacio Bono Parisi, Inés Pacharoni

The Matrix Bochner Problem aims to classify which weight matrices have their sequence of orthogonal polynomials as eigenfunctions of a second-order differential operator. Casper and Yakimov, in [4], demonstrated that, under certain hypotheses, all solutions to the Matrix Bochner Problem are noncommutative bispectral Darboux transformations of a direct sum of classical scalar weights. This paper aims to provide the first proof that there are solutions to the Matrix Bochner Problem that do not arise through a noncommutative bispectral Darboux transformation of any direct sum of classical scalar weights. This initial example could contribute to a more comprehensive understanding of the general solution to the Matrix Bochner Problem.

矩阵波赫纳问题旨在分类哪些权重矩阵的正交多项式序列是二阶微分算子的特征函数。Casper 和 Yakimov 在 [4] 中证明，在某些假设条件下，矩阵波赫纳问题的所有解都是经典标量权重直接和的非交换双谱达尔布克斯变换。本文旨在首次证明，矩阵波赫纳问题的解并不是通过经典标量权重直接和的非交换双谱达尔布克斯变换产生的。这个初步例子有助于更全面地理解矩阵波赫纳问题的一般解法。

引用次数: 0

Minimal projections onto subspaces generated by sign-matrices 符号矩阵生成的子空间上的最小投影

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2024-12-01 Epub Date: 2024-07-26 DOI: 10.1016/j.jat.2024.106084

Beata Derȩgowska , Barbara Lewandowska , Grzegorz Lewicki

The aim of this paper is to calculate relative and absolute projection constants of certain subspaces of $l_{1}^{(n)}$ and $l_{\infty}^{(n)}$ generated by eigenvectors of sign matrices. The main tool in our considerations is so called Chalmers–Metcalf operator (see Chalmers & Metcalf (1994) and Lewicki & Prophet (2021)). Also, some results from Castejon & Lewicki (2014) and Castejon & Lewicki (2019) will be applied.

本文旨在计算由符号矩阵特征向量生成的 l1(n) 和 l∞(n) 的某些子空间的相对和绝对投影常数。我们考虑的主要工具是所谓的 Chalmers-Metcalf 算子（见 Chalmers & Metcalf (1994) 和 Lewicki & Prophet (2021)）。此外，我们还将应用 Castejon & Lewicki (2014) 和 Castejon & Lewicki (2019) 的一些结果。

引用次数: 0

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Journal of Approximation Theory

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