Journal of Approximation Theory最新文献

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On the representability of a continuous multivariate function by sums of ridge functions 论脊函数之和对连续多元函数的可表示性

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2024-10-09 DOI: 10.1016/j.jat.2024.106105

Rashid A. Aliev , Fidan M. Isgandarli

In this paper, new conditions are found for the representability of a continuous multivariate function as a sum of ridge functions. Using these conditions, we give a new proof for the earlier theorem solving the problem, posed by A.Pinkus in his monograph “Ridge Functions”, up to a multivariate polynomial. That is, we show that if a continuous multivariate function has a representation as a sum of arbitrarily behaved ridge functions, then it can be represented as a sum of continuous ridge functions and some multivariate polynomial.

本文为连续多元函数作为脊函数之和的可表示性找到了新的条件。利用这些条件，我们对 A.Pinkus 在他的专著《脊函数》中提出的解决这一问题的早期定理给出了新的证明，直至多元多项式。也就是说，我们证明了如果一个连续多元函数可以表示为任意表现的脊函数之和，那么它就可以表示为连续脊函数与某个多元多项式之和。

引用次数: 0

On sharp heat kernel estimates in the context of Fourier–Dini expansions 关于傅立叶-迪尼展开中的尖锐热核估计值

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2024-09-28 DOI: 10.1016/j.jat.2024.106103

Bartosz Langowski , Adam Nowak

We prove sharp estimates of the heat kernel associated with Fourier–Dini expansions on

(0, 1)

equipped with Lebesgue measure and the Neumann condition imposed on the right endpoint. Then we give several applications of this result including sharp bounds for the corresponding Poisson and potential kernels, sharp mapping properties of the maximal heat semigroup and potential operators and boundary convergence of the Fourier–Dini semigroup.

我们证明了与（0,1）上的傅里叶-迪尼展开相关的热核的尖锐估计值，该热核配有勒贝格度量和施加于右端点的诺伊曼条件。然后，我们给出了这一结果的若干应用，包括相应泊松核和势核的尖锐边界、最大热半群和势算子的尖锐映射性质以及傅里叶-迪尼半群的边界收敛。

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 : 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 : 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 Bernstein- and Marcinkiewicz-type inequalities on multivariate Cα-domains 论多变量 Cα 域上的伯恩斯坦和马钦凯维奇型不等式

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2024-09-16 DOI: 10.1016/j.jat.2024.106101

Feng Dai , András Kroó , Andriy Prymak

We prove new Bernstein and Markov type inequalities in $L^{p}$ spaces associated with the normal and the tangential derivatives on the boundary of a general compact $C^{α}$ -domain with $1 \leq α \leq 2$ . These estimates are also applied to establish Marcinkiewicz type inequalities for discretization of $L^{p}$ norms of algebraic polynomials on $C^{α}$ -domains with asymptotically optimal number of function samples used.

我们证明了 Lp 空间中与 1≤α≤2 的一般紧凑 Cα 域边界上的法导数和切导数相关的新伯恩斯坦和马尔可夫式不等式。这些估计值还被应用于建立 Marcinkiewicz 型不等式，用于 Cα 域上代数多项式 Lp 准则的离散化，并使用渐近最优的函数样本数。

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 : 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

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 : 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

Dual spaces vs. Haar measures of polynomial hypergroups 多项式超群的对偶空间与哈氏度量

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2024-09-07 DOI: 10.1016/j.jat.2024.106099

Stefan Kahler , Ryszard Szwarc

Many symmetric orthogonal polynomials

{(P_{n} (x))}_{n \in N_{0}}

induce a hypergroup structure on

N_{0}

. The Haar measure is the counting measure weighted with

h (n) ≔ 1 / \int_{R} P_{n}^{2} (x) d μ (x) \geq 1

, where

μ

denotes the orthogonalization measure. We observed that many naturally occurring examples satisfy the remarkable property

h (n) \geq 2 (n \in N)

. We give sufficient criteria and particularly show that

h (n) \geq 2 (n \in N)

if the (Hermitian) dual space

\hat{N_{0}}

equals the full interval

[- 1, 1]

, which is fulfilled by an abundance of examples. We also study the role of nonnegative linearization of products (and of the resulting harmonic and functional analysis). Moreover, we construct two example types with

h (1) < 2

. To our knowledge, these are the first such examples. The first type is based on Karlin–McGregor polynomials, and

\hat{N_{0}}

consists of two intervals and can be chosen “maximal” in some sense;

h

is of quadratic growth. The second type relies on hypergroups of strong compact type;

h

grows exponentially, and

\hat{N_{0}}

is discrete.

许多对称正交多项式 (Pn(x))n∈N0 都会在 N0 上诱发超群结构。哈氏度量是以 h(n)≔1/∫RPn2(x)dμ(x)≥1 加权的计数度量，其中 μ 表示正交化度量。我们观察到许多自然出现的例子都满足 h(n)≥2(n∈N) 这一显著特性。我们给出了充分的标准，并特别表明，如果（赫米特）对偶空间 N0 ̂ 等于整个区间 [-1,1]，则 h(n)≥2(n∈N)。我们还研究了乘积的非负线性化（以及由此产生的谐波分析和函数分析）的作用。此外，我们还构建了两个 h(1)<2 的示例类型。第一种类型基于 Karlin-McGregor 多项式，N0̂ 由两个区间组成，在某种意义上可以选择 "最大"；h 为二次增长。第二种类型依赖于强紧凑类型的超群；h 以指数形式增长，而 N0 ̂ 是离散的。

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 : 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 $0 0$ 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

Distribution of the zeros of polynomials near the unit circle 单位圆附近多项式零点的分布

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2024-08-08 DOI: 10.1016/j.jat.2024.106087

Mithun Kumar Das

We estimate the number of zeros of a polynomial in $ℂ [z]$ within any small circular disk centered on the unit circle, which improves and comprehensively extends a result established by Borwein, Erdélyi, and Littmann in 2008. Furthermore, by combining this result with Euclidean geometry, we derive an upper bound on the number of zeros of such a polynomial within a region resembling a gear wheel. Additionally, we obtain a sharp upper bound on the annular discrepancy of such zeros near the unit circle. Our approach builds upon a modified version of the method described in Borwein et al. (2008), combined with the refined version of the best-known upper bound for angular discrepancy of zeros of polynomials.

我们估算了ℂ[z]多项式在以单位圆为中心的任何小圆盘内的零点数，这改进并全面扩展了博尔文、埃尔德利和利特曼在 2008 年建立的一个结果。此外，通过将这一结果与欧几里得几何相结合，我们推导出了在类似齿轮的区域内该多项式的零点个数上限。此外，我们还获得了单位圆附近此类零点环差的尖锐上界。我们的方法建立在 Borwein 等人（2008 年）所描述方法的改进版基础之上，并结合了多项式零点角度差异的最著名上界的改进版。

引用次数: 0

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