Journal of Approximation Theory最新文献

英文中文

Optimal heat kernel bounds and asymptotics on Damek–Ricci spaces

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-01-11 DOI: 10.1016/j.jat.2025.106144

Tommaso Bruno , Federico Santagati

We give optimal bounds for the radial, space and time derivatives of arbitrary order of the heat kernel of the Laplace–Beltrami operator on Damek–Ricci spaces. In the case of symmetric spaces of rank one, these complete and actually improve conjectured estimates by Anker and Ji. We also provide asymptotics at infinity of all the radial and time derivates of the kernel. Along the way, we provide sharp bounds for all the derivatives of the Riemannian distance and obtain analogous bounds for those of the heat kernel of the distinguished Laplacian.

引用次数: 0

Bounds for extreme zeros of Meixner–Pollaczek polynomials

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2024-12-21 DOI: 10.1016/j.jat.2024.106142

A.S. Jooste , K. Jordaan

In this paper we consider connection formulae for orthogonal polynomials in the context of Christoffel transformations for the case where a weight function, not necessarily even, is multiplied by an even function

c_{2 k} (x), k \in N_{0}

, to determine new lower bounds for the largest zero and upper bounds for the smallest zero of a Meixner–Pollaczek polynomial. When

p_{n}

is orthogonal with respect to a weight

w (x)

and

g_{n - m}

is orthogonal with respect to the weight

c_{2 k} (x) w (x)

, we show that

k \in {0, 1, \dots, m}

is a necessary and sufficient condition for existence of a connection formula involving a polynomial

G_{m - 1}

of degree

(m - 1)

, such that the

(n - 1)

zeros of

G_{m - 1} g_{n - m}

and the

n

zeros of

p_{n}

interlace. We analyse the new inner bounds for the extreme zeros of Meixner–Pollaczek polynomials to determine which bounds are the sharpest. We also briefly discuss bounds for the zeros of Pseudo-Jacobi polynomials.

引用次数: 0

Twenty-five years of greedy bases

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2024-12-21 DOI: 10.1016/j.jat.2024.106141

Fernando Albiac , José L. Ansorena , Vladimir Temlyakov

Although the basic idea behind the concept of a greedy basis had been around for some time, the formal development of a theory of greedy bases was initiated in 1999 with the publication of the article [S.V. Konyagin and V.N. Temlyakov, A remark on greedy approximation in Banach spaces, East J. Approx. 5 (3) (1999), 365-379]. The theoretical simplicity of the thresholding greedy algorithm became a model for a procedure widely used in numerical applications and the subject of greedy bases evolved very rapidly from the point of view of approximation theory. The idea of studying greedy bases and related greedy algorithms attracted also the attention of researchers with a classical Banach space theory background. From the more abstract point of functional analysis, the theory of greedy bases and its derivates evolved very fast as many fundamental results were discovered and new ramifications branched out. Hundreds of papers on greedy-like bases and several monographs have been written since the appearance of the aforementioned foundational paper. After twenty-five years, the theory is very much alive and it continues to be a very active research topic both for functional analysts and for researchers interested in the applied nature of nonlinear approximation alike. This is why we believe it is a good moment to gather a selection of 25 open problems (one per year since 1999!) whose solution would contribute to advance the state of art of this beautiful topic.

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引用次数: 0

Improved Stein inequalities for the Fourier transform 改进的傅立叶变换斯坦因不等式

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2024-11-22 DOI: 10.1016/j.jat.2024.106126

Erlan D. Nursultanov , Durvudkhan Suragan

In this paper, we present a refined version of the (classical) Stein inequality for the Fourier transform, elevating it to a new level of accuracy. Furthermore, we establish extended analogues of a more precise version of the Stein inequality for the Fourier transform, broadening its applicability from the range

1 < p < 2

2 \leq p < \infty

在本文中，我们提出了傅立叶变换的（经典）斯坦因不等式的改进版，将其精确度提升到了一个新的水平。此外，我们还建立了更精确版本的傅立叶变换斯坦因不等式的扩展类比，将其适用范围从 1<p<2 扩大到 2≤p<∞。

引用次数: 0

Spherical basis functions in Hardy spaces with localization constraints 具有定位约束条件的哈代空间中的球面基函数

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2024-11-21 DOI: 10.1016/j.jat.2024.106124

C. Gerhards, X. Huang

Subspaces obtained by the orthogonal projection of locally supported square-integrable vector fields onto the Hardy spaces

H_{+} (S)

and

H_{-} (S)

, respectively, play a role in various inverse potential field problems since they characterize the uniquely recoverable components of the underlying sources. Here, we consider approximation in these subspaces by a particular set of spherical basis functions. Error bounds are provided along with further considerations on norm-minimizing vector fields that satisfy the underlying localization constraint. The new aspect here is that the used spherical basis functions are themselves members of the subspaces under consideration.

由局部支持的方整矢量场分别正交投影到哈代空间 H+(S)和 H-(S)而得到的子空间，在各种反势场问题中发挥着作用，因为它们描述了基础源的唯一可恢复成分。在此，我们考虑用一组特定的球面基函数来逼近这些子空间。在提供误差边界的同时，我们还进一步考虑了满足基本定位约束条件的规范最小化矢量场。这里的新颖之处在于，所使用的球面基函数本身就是所考虑的子空间的成员。

引用次数: 0

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

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