Journal of Approximation Theory最新文献

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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

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

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

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 : 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

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 : 2024-10-09 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 : 2024-10-09 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

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