Journal of Approximation Theory最新文献

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Randomized approximation of summable sequences — adaptive and non-adaptive 可求和序列的随机逼近--适应性和非适应性

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2024-12-01 Epub Date: 2024-06-12 DOI: 10.1016/j.jat.2024.106056

Robert J. Kunsch , Erich Novak , Marcin Wnuk

We prove lower bounds for the randomized approximation of the embedding $ℓ_{1}^{m} ↪ ℓ_{\infty}^{m}$ based on algorithms that use arbitrary linear (hence non-adaptive) information provided by a (randomized) measurement matrix $N \in R^{n \times m}$ . These lower bounds reflect the increasing difficulty of the problem for $m \to \infty$ , namely, a term $\sqrt{log m}$ in the complexity $n$ . This result implies that non-compact operators between arbitrary Banach spaces are not approximable using non-adaptive Monte Carlo methods. We also compare these lower bounds for non-adaptive methods with upper bounds based on adaptive, randomized methods for recovery for which the complexity $n$ only exhibits a $(log log m)$ -dependence. In doing so we give an example of linear problems where the error for adaptive vs. non-adaptive Monte Carlo methods shows a gap of order $n^{1 / 2} {(log n)}^{- 1 / 2}$ .

我们证明了基于使用由（随机）测量矩阵提供的任意线性（因此非适应性）信息的算法的嵌入随机近似的下限。这些下界反映了问题难度的增加，即复杂度中的一个项......。这一结果意味着，任意巴拿赫空间之间的非紧凑算子无法用非自适应蒙特卡洛方法逼近。我们还将这些非自适应方法的下界与基于自适应随机方法的上界进行了比较，对于后者，复杂度只表现出-依赖关系。为此，我们举了一个线性问题的例子，在这些问题中，自适应蒙特卡洛方法与非自适应蒙特卡洛方法的误差显示出数量级为.的差距。

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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 : 2024-12-01 Epub 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

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 : 2024-12-01 Epub 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

Barycentric rational interpolation method for solving 3 dimensional convection–diffusion equation 求解三维对流扩散方程的巴利心理性插值法

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2024-12-01 Epub Date: 2024-10-09 DOI: 10.1016/j.jat.2024.106106

Jin Li, Yongling Cheng

Barycentric rational interpolation collocation method (BRICM) is presented to solve 3-dimensional convection–diffusion (CD) equation. The unknown value is approximated by barycentric rational interpolation basis, the discrete CD equation is written into the matrix equation. At last, the stability and convergence rate of BRIM for CD equation are proven and a numerical example is illustrated in our results.

提出了用于求解三维对流扩散（CD）方程的重心有理插值法（BRICM）。未知值由重心有理插值基近似，离散 CD 方程被写入矩阵方程。最后，证明了对流扩散方程 BRIM 的稳定性和收敛率，并以数值结果为例进行了说明。

引用次数: 0

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 : 2024-12-01 Epub 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

Convergence in distribution of the Bernstein–Durrmeyer kernel and pointwise convergence of a generalised operator for functions of bounded variation 伯恩斯坦-达尔迈耶核分布的收敛性和有界变化函数广义算子的点收敛性

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2024-12-01 Epub Date: 2024-08-08 DOI: 10.1016/j.jat.2024.106086

Mohammed Taariq Mowzer

We study the convergence of Bernstein type operators leading to two results. The first: The kernel $K_{n}$ of the Bernstein–Durrmeyer operator at each point $x \in (0, 1)$ — that is $K_{n} (x, t) d t$ — once standardised converges to the normal distribution. The second result computes the pointwise limit of a generalised Bernstein–Durrmeyer operator applied to — possibly discontinuous — functions $f$ of bounded variation.

我们对伯恩斯坦型算子的收敛性进行了研究，得出了两个结果。第一个结果：伯恩斯坦-杜尔迈耶算子在每一点 x∈(0,1) 的核 Kn（即 Kn(x,t)dt）一旦标准化，就会收敛于正态分布。第二个结果是计算应用于有界变化函数 f（可能不连续）的广义伯恩斯坦-德尔迈尔算子的点极限。

引用次数: 0

On approximation by rational functions in Musielak–Orlicz spaces 论穆西拉克-奥利兹空间中有理函数的逼近

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.2024.106083

Wojciech M. Kozlowski , Gianluca Vinti

We consider best approximation by rational functions in Musielak–Orlicz spaces of real-valued measurable functions over the unit interval equipped with the Lebesgue measure. We prove several properties of the respective multi-value projection operator, including its semi-continuity. Our results generalise known results for Lebesgue and variable Lebesgues spaces, and can be applied to special cases including Orlicz spaces and variable Lebesgue spaces with weights. We touch upon applications to image processing.

我们考虑的是在单位区间上的实值可测函数的 Musielak-Orlicz 空间中，用有理函数进行最佳逼近，并配备 Lebesgue 度量。我们证明了相应多值投影算子的几个性质，包括其半连续性。我们的结果概括了已知的 Lebesgue 和可变 Lebesgues 空间的结果，并可应用于特殊情况，包括 Orlicz 空间和有权重的可变 Lebesgue 空间。我们还谈到了在图像处理中的应用。

引用次数: 0

The Machado–Bishop theorem in the uniform topology 统一拓扑中的马查多-毕夏普定理

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.2024.106085

Deliang Chen

The Machado–Bishop theorem for weighted vector-valued functions vanishing at infinity has been extensively studied. In this paper, we give an analogue of Machado’s distance formula for bounded weighted vector-valued functions. A number of applications are given; in particular, some types of the Bishop–Stone–Weierstrass theorem for bounded vector-valued continuous spaces in the uniform topology are discussed; the splitting of $C (I \times J, X \otimes Y)$ as the closure of $C (I, X) \otimes C (J, Y)$ in different senses and the extension of continuous vector-valued functions are studied.

对于在无穷大处消失的加权矢量值函数，马查多-毕夏普定理已被广泛研究。本文给出了有界加权向量值函数的马查多距离公式。本文给出了一些应用；特别是讨论了均匀拓扑中有界向量值连续空间的 Bishop-Stone-Weierstrass 定理的一些类型；研究了作为 C(I,X)⊗C(J,Y) 闭包的 C(I×J,X⊗Y) 在不同意义上的分裂以及连续向量值函数的扩展。

引用次数: 0

In memory of Peter Borwein 纪念彼得-博尔文

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2024-11-01 Epub Date: 2024-07-05 DOI: 10.1016/j.jat.2024.106075

Boris Shekhtman

引用次数: 0

Peter Borwein: A philosopher’s mathematician 彼得-博文哲学家的数学家

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2024-11-01 Epub Date: 2024-07-05 DOI: 10.1016/j.jat.2024.106071

David H. Bailey

引用次数: 0

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