Journal of Approximation Theory最新文献

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Yosida approximations of the cumulative distribution function and applications in survival analysis 累积分布函数的约西达近似值及其在生存分析中的应用

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2024-11-21 DOI: 10.1016/j.jat.2024.106123

Miroslav Bačák

The Yosida approximation method is a classic regularization technique in maximal monotone operator theory. In the present paper, however, we apply it to the cumulative distribution function (cdf) and study its properties in the context of statistics. In that case the Yosida approximation transforms a given cdf into a new cdf with better continuity properties, namely the new cdf is Lipschitz continuous, and its distance to the original cdf as well as its Lipschitz constant are both controlled by a parameter.

When applied to an empirical cdf, which is arguably the most important case in practice, the Yosida approximation yields a continuous piecewise linear cdf in a systematic way, underpinned by a versatile theoretical framework. This provides a new smoothing technique which to our knowledge has not been explored in the literature yet.

After establishing several theoretical statistical properties of Yosida approximations we show possible applications to survival analysis. Finally, we pose two open problems in order to stimulate further research along these lines.

Yosida 近似方法是最大单调算子理论中的一种经典正则化技术。但在本文中，我们将其应用于累积分布函数（cdf），并研究其在统计学中的特性。在这种情况下，约西达近似将给定的 cdf 转换为具有更好连续性的新 cdf，即新的 cdf 是 Lipschitz 连续的，它与原始 cdf 的距离以及 Lipschitz 常量都由一个参数控制。当应用于经验 cdf（这可以说是实践中最重要的情况）时，约西达近似以一种系统的方式产生了连续的片断线性 cdf，并以一个通用的理论框架为基础。这提供了一种新的平滑技术，据我们所知，该技术尚未在文献中得到探讨。在建立了约西达近似的几个理论统计特性后，我们展示了在生存分析中的可能应用。最后，我们提出了两个有待解决的问题，以激励沿着这些思路开展进一步的研究。

引用次数: 0

Classification of 2-orthogonal polynomials with Brenke type generating functions 具有布伦克型生成函数的二正交多项式的分类

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2024-11-21 DOI: 10.1016/j.jat.2024.106125

Hamza Chaggara , Abdelhamid Gahami

The Brenke type generating functions are the polynomial generating functions of the form

\sum_{n = 0}^{\infty} \frac{P_{n} (x)}{n!} t^{n} = A (t) B (x t),

where

A

and

B

are two formal power series subject to the conditions

A (0) \cdot B^{(k)} (0) \neq 0, k = 0, 1, 2, \dots

. In this work, we shall determine all Brenke-type polynomials when they are also 2-orthogonal polynomial sequences, that is to say, polynomials with Brenke type generating function and satisfying one standard four-term recurrence relation. That allows us, on one hand, to obtain new 2-orthogonal sequences generalizing known orthogonal families of polynomials, and on the other hand, to recover particular cases of polynomial sequences discovered in the context of

d

-orthogonality.

The classification is based on the discussion of a three-order difference equation induced by the four-term recurrence relation satisfied by the considered polynomials. This study is motivated by the work of Chihara (1968) who gave all pairs

(A (t), B (t))

for which

{P_{n} (x)}_{n \geq 0}

is an orthogonal polynomial sequence. In some cases, we give the expression of the moments associated to the two-dimensional functional of orthogonality.

布伦克型生成函数是形式为 ∑n=0∞Pn(x)n!tn=A(t)B(xt) 的多项式生成函数，其中 A 和 B 是两个形式幂级数，条件是 A(0)⋅B(k)(0)≠0,k=0,1,2,....。在这项工作中，我们将确定所有布伦克型多项式，当它们也是 2 正交多项式序列时，也就是说，具有布伦克型生成函数并满足一个标准四项递推关系的多项式。这使得我们一方面可以获得新的二正交序列，概括已知的多项式正交族，另一方面可以恢复在 d 正交背景下发现的多项式序列的特殊情况。千原（Chihara，1968 年）给出了{Pn(x)}n≥0 为正交多项式序列的所有对 (A(t),B(t))。在某些情况下，我们给出了与正交性二维函数相关的矩的表达式。

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引用次数: 0

Extensions of the Bloch-Pólya theorem on the number of real zeros of polynomials (II) 关于多项式实零点个数的布洛赫-波利亚定理的扩展 (II)

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2024-11-15 DOI: 10.1016/j.jat.2024.106122

Tamás Erdélyi

We prove that there is an absolute constant

c > 0

such that for every

a_{0}, a_{1}, \dots, a_{n} \in [1, M], 1 \leq M \leq \frac{1}{4} exp (\frac{n}{9}),

there are

b_{0}, b_{1}, \dots, b_{n} \in {- 1, 0, 1}

such that the polynomial

P

of the form

P (z) = \sum_{j = 0}^{n} b_{j} a_{j} z^{j}

has at least

c {(\frac{n}{log (4 M)})}^{1 / 2} - 1

distinct sign changes in

I_{a} : = (1 - 2 a, 1 - a)

, where

a : = {(\frac{log (4 M)}{n})}^{1 / 2} \leq 1 / 3

. This improves and extends earlier results of Bloch and Pólya and Erdélyi and, as a special case, recaptures a special case of a more general recent result of Jacob and Nazarov.

我们证明存在一个绝对常数 c>0，使得对于每一个 a0,a1,...,an∈[1,M],1≤M≤14expn9,有 b0,b1,...,bn∈{-1,0,1}，使得形式为 P(z)=∑j=0nbjajzj 的多项式 P 在 Ia 中至少有 cnlog(4M)1/2-1 个不同的符号变化：=(1-2a,1-a)，其中 a:=log(4M)n1/2≤1/3.这改进并扩展了布洛赫、波利亚和埃尔德利的早期结果，并作为一个特例，重现了雅各布和纳扎罗夫的一个更普遍的最新结果的特例。

引用次数: 0

Random sampling and polynomial-free interpolation by Generalized MultiQuadrics 广义多二次方的随机抽样和无多项式插值法

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2024-11-15 DOI: 10.1016/j.jat.2024.106119

A. Sommariva, M. Vianello

We prove that interpolation matrices for Generalized MultiQuadrics (GMQ) of order greater than one are almost surely nonsingular without polynomial addition, in any dimension and with any continuous random distribution of sampling points. We also include a new class of generalized MultiQuadrics recently proposed by Buhmann and Ortmann.

我们证明，在任何维度和任何连续随机分布的采样点上，阶数大于 1 的广义多曲（GMQ）的插值矩阵几乎肯定是非奇异的，不需要多项式加法。我们还包括了布曼和奥特曼最近提出的一类新的广义多曲。

引用次数: 0

Strictly positive definite functions on spheres 球面上的严格正定函数

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2024-11-15 DOI: 10.1016/j.jat.2024.106120

Tianshi Lu

In this paper, we proved that a positive definite radial function on

R^{d}

with support in

[0, π]

is strictly positive definite on the sphere

S^{d}

and real projective space

{RP}^{d}

for odd

d \geq 3

. We also proved that the truncated power function

{(t - \cdot)}_{+}^{(d + 1) / 2}

is strictly positive definite on

S^{d}

and

{RP}^{d}

for

d \geq 2

and

t \in (0, π]

在本文中，我们证明了对于奇数 d≥3 时，Rd 上支持在 [0,π] 中的正定径向函数在球面 Sd 和实投影空间 RPd 上是严格正定的。我们还证明，对于 d≥2 和 t∈(0,π]，截断幂函数 (t-⋅)+(d+1)/2 在 Sd 和 RPd 上是严格正定的。

引用次数: 0

Generalized Bell polynomials 广义贝尔多项式

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2024-11-15 DOI: 10.1016/j.jat.2024.106121

Antonio J. Durán

In this paper, generalized Bell polynomials

{(b_{n}^{ϕ})}_{n}

associated to a sequence of real numbers

ϕ = {(ϕ_{i})}_{i = 1}^{\infty}

are introduced. Bell polynomials correspond to

ϕ_{i} = 0

i \geq 1

. We prove that when

ϕ_{i} \geq 0

i \geq 1

: (a) the zeros of the generalized Bell polynomial

b_{n}^{ϕ}

are simple, real and non positive; (b) the zeros of

b_{n + 1}^{ϕ}

interlace the zeros of

b_{n}^{ϕ}

; (c) the zeros are decreasing functions of the parameters

ϕ_{i}

. We find a hypergeometric representation for the generalized Bell polynomials. As a consequence, it is proved that the class of all generalized Bell polynomials is actually the same class as that of all Laguerre multiple polynomials of the first kind.

本文介绍了与实数序列 ϕ=(ϕi)i=1∞ 相关的广义贝尔多项式 (bnj)n。我们证明了当ϕi≥0，i≥1 时：（a）广义贝尔多项式 bnj 的零点是简单、实且非正的；（b）bn+1j 的零点与 bnj 的零点交错；（c）零点是参数ϕi 的递减函数。我们找到了广义贝尔多项式的超几何表示。因此，我们证明了所有广义贝尔多项式的类别实际上与所有拉盖尔第一类多重多项式的类别相同。

引用次数: 0

Sampling theorems with derivatives in shift-invariant spaces generated by periodic exponential B-splines 周期性指数 B 样条生成的移位不变空间中的导数采样定理

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2024-11-15 DOI: 10.1016/j.jat.2024.106118

Karlheinz Gröchenig, Irina Shafkulovska

We derive sufficient conditions for sampling with derivatives in shift-invariant spaces generated by a periodic exponential B-spline. The sufficient conditions are expressed with a new notion of measuring the gap between consecutive samples. These conditions are near optimal, and, in particular, they imply the existence of sampling sets with lower Beurling density arbitrarily close to the necessary density.

我们推导出了在由周期性指数 B-样条生成的移变空间中进行导数采样的充分条件。这些充分条件用一个测量连续采样间隙的新概念来表示。这些条件接近最优，特别是，它们意味着存在任意接近必要密度的下贝林密度采样集。

引用次数: 0

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 : 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

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 : 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

In search of a higher Bochner theorem 寻找更高的波赫纳定理

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2024-10-22 DOI: 10.1016/j.jat.2024.106114

Emil Horozov , Boris Shapiro , Miloš Tater

We initiate the study of a natural generalisation of the classical Bochner–Krall problem asking which linear ordinary differential operators possess sequences of eigenpolynomials satisfying linear recurrence relations of finite length; the classical case corresponds to the 3-term recurrence relations with real coefficients subject to some extra restrictions. We formulate a general conjecture and prove it in the first non-trivial case of operators of order 3.

我们开始研究经典 Bochner-Krall 问题的自然广义化，即线性常微分算子具有满足有限长度线性递推关系的特征多项式序列；经典情况对应于受一些额外限制的实系数 3 期递推关系。我们提出了一个一般性猜想，并在第一个阶数为 3 的算子的非难情形中证明了这一猜想。

引用次数: 0

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