Journal of Approximation Theory最新文献

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Embeddings of block-radial functions — approximation properties and nuclearity 块径向函数的嵌入。近似性质和核

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-07-23 DOI: 10.1016/j.jat.2025.106214

Alicja Dota , Leszek Skrzypczak

Let

R_{γ} B_{p, q}^{s} (R^{d})

be a subspace of the Besov space

B_{p, q}^{s} (R^{d})

that consists of block-radial (multi-radial) functions. We study the asymptotic behaviour of approximation numbers of compact embeddings

id : R_{γ} B_{p_{1}, q_{1}}^{s_{1}} (R^{d}) \to R_{γ} B_{p_{2}, q_{2}}^{s_{2}} (R^{d})

. Moreover, we find a sufficient and necessary condition for nuclearity of the above embeddings. Analogous results are proved for fractional Sobolev spaces

R_{γ} H_{p}^{s} (R^{d})

设RγBp，qs（Rd）是Besov空间Bp，qs（Rd）的一个子空间，它由块径向（多径向）函数组成。研究了紧嵌入id:RγBp1，q1s1（Rd）→RγBp2,q2s2（Rd）的逼近数的渐近性。此外，我们还发现了上述嵌入具有核性的充分必要条件。证明了分数Sobolev空间RγHps（Rd）的类似结果。

引用次数: 0

Best approximations and their extensions in Lorentz Gamma spaces 洛伦兹空间中的最佳逼近及其扩展

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-07-08 DOI: 10.1016/j.jat.2025.106212

F.D. Kovac , F.E. Levis , L. Zabala

In this article, we investigate the best approximation operator from a finite-dimensional linear space defined on Lorentz Gamma spaces

Γ_{w, p}

for

1 \leq p < \infty

. We extend the best approximation operator from

Γ_{w, p}

to the larger space

Γ_{w, p - 1}

and establish several key properties of these operators.

在本文中，我们研究了在Lorentz Gamma空间Γw上定义的有限维线性空间中的最佳逼近算子，p为1≤p<；∞。我们将最佳逼近算子从Γw，p推广到更大的空间Γw，p−1，并建立了这些算子的几个关键性质。

引用次数: 0

Matrix valued orthogonal polynomials arising from hexagon tilings with 3 × 3-periodic weightings 由3 × 3周期加权的六边形平铺引起的矩阵值正交多项式

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-05-26 DOI: 10.1016/j.jat.2025.106202

Arno B.J. Kuijlaars

Matrix valued orthogonal polynomials (MVOP) appear in the study of doubly periodic tiling models. Of particular interest is their limiting behavior as the degree tends to infinity. In recent years, MVOP associated with doubly periodic domino tilings of the Aztec diamond have been successfully analyzed. The MVOP related to doubly periodic lozenge tilings of a hexagon are more complicated. In this paper we focus on a special subclass of hexagon tilings with 3 × 3 periodicity. The special subclass leads to a genus one spectral curve with additional symmetries that allow us to find an equilibrium measure in an external field explicitly. The equilibrium measure gives the asymptotic distribution for the zeros of the determinant of the MVOP. The associated

g

-functions appear in the strong asymptotic formula for the MVOP that we obtain from a steepest descent analysis of the Riemann–Hilbert problem for MVOP.

矩阵值正交多项式（MVOP）出现在双周期平铺模型的研究中。特别令人感兴趣的是当阶趋于无穷大时它们的极限行为。近年来，对阿兹特克钻石双周期多米诺骨牌铺层的MVOP进行了成功的分析。六边形双周期菱形平铺的MVOP更为复杂。本文研究了具有3 × 3周期性的六边形平铺的一个特殊子类。这个特殊的子类导致了一个具有额外对称性的谱曲线，使我们能够明确地在一个外场中找到一个平衡测度。平衡测度给出了MVOP的行列式零点的渐近分布。相关的g函数出现在MVOP的强渐近公式中，该公式是我们从MVOP的Riemann-Hilbert问题的最陡下降分析中得到的。

引用次数: 0

Intrinsic interpolation, near-circularity and maximal convergence 内禀插值，近圆度和最大收敛

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-05-24 DOI: 10.1016/j.jat.2025.106201

Hans-Peter Blatt

Let

E

be compact and connected with

cap E > 0

and connected complement

Ω = \bar{ℂ} ∖ E

, let

g_{Ω} (z, \infty)

be the Green’s function of

Ω

with pole at infinity and let

E_{σ} ≔ {z \in Ω : g_{Ω} (z, \infty) < log σ} \cup E, 1 < σ < \infty,

be the Green domains with boundaries

Γ_{σ}

. Let

f

be holomorphic on

E

and let

ρ (f)

denote the maximal parameter of holomorphy of

f

and let

{(p_{n})}_{n \in N}

be a sequence of polynomials converging maximally to

f

E

. If

σ

1 < σ < ρ (f) < \infty

, is fixed and if

m_{n} (σ)

denotes the number of interpolation points of

p_{n}

f

E_{σ}

with normalized counting measure

μ_{σ, n}

, then there exists a subset

Λ \subset N

such that

m_{n} (σ) = n + o (n) as n \in Λ, n \to \infty,

设E是紧致的，并且与capE>；0和连通补Ω=¯∈E相连，设gΩ（z,∞）是Ω的极点在无穷远处的Green函数，设Eσ∈Ω：gΩ（z,∞）<logσ}∪E,1<σ<；∞是有边界的Green域Γσ。让f E和上全纯让ρ(f)表示最大的正则参数f对所测试,让∈N是一个多项式序列收敛最大f E .如果σ,1 & lt;σ& lt;ρ(f) & lt;∞,是固定的,如果mn(σ)表示pn的数量的插值点与规范化计数测量μf Eσσ,N,那么存在一个子集Λ⊂N, mn(σ)= N + o (N) asn∈Λ,N→∞,μσ,N | E +μσ,N |Ω⟶∗μEasn∈Λ,N→∞,在μσ,N =μσ,N | E +μσ,N |Ω,μσ，n|Ê表示μσ的平衡测度，n|E在E的边界上，μE是E的平衡测度，并且存在一个收敛于σ的序列σnn∈Λ，使得闭合曲线γn=(f−pn)（Γσn）不经过0点，圈数Indγn(0)满足Indγn(0)=mn(σn)=n+o(n)asn∈Λ，n→∞。

引用次数: 0

Bernstein-type inequalities for mean n-valent functions 平均n价函数的bernstein型不等式

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-05-16 DOI: 10.1016/j.jat.2025.106200

Anton Baranov , Ilgiz Kayumov , Rachid Zarouf

We derive new integral estimates of the derivatives of mean

n

-valent functions in the unit disc. Our results develop and complement estimates obtained by E. P. Dolzhenko and A. A. Pekarskii, as well as recent inequalities obtained by the authors. As an application, we improve some inverse theorems of rational approximation due to Dolzhenko.

给出了单位圆盘中n价平均函数导数的新的积分估计。我们的结果发展和补充了E. P. Dolzhenko和A. A. Pekarskii的估计，以及作者最近得到的不等式。作为应用，我们改进了一些由于Dolzhenko的有理逼近的逆定理。

引用次数: 0

Universal discretization and sparse recovery 通用离散化和稀疏恢复

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-05-13 DOI: 10.1016/j.jat.2025.106199

F. Dai , V. Temlyakov

Recently, it was discovered that for a given function class

F

the error of best linear recovery in the square norm can be bounded above by the Kolmogorov width of

F

in the uniform norm. That analysis is based on deep results in discretization of the square norm of functions from finite-dimensional subspaces. In this paper we show how very recent results on universal discretization of the square norm of functions from a collection of finite-dimensional subspaces lead to a Lebesgue-type inequality between the error of sparse recovery in the square norm provided by the algorithm based on least squares operator and best sparse approximations in the uniform norm with respect to appropriate dictionaries.

最近，我们发现对于给定的函数类F，在平方范数中最佳线性恢复的误差可以以F在一致范数中的Kolmogorov宽度为界。该分析是基于有限维子空间中函数的平方范数离散化的深入结果。在本文中，我们展示了最近关于有限维子空间集合的函数的平方范数的普遍离散化的结果如何导致基于最小二乘算子的算法提供的平方范数的稀疏恢复误差与关于适当字典的均匀范数的最佳稀疏近似之间的lebesgue型不等式。

引用次数: 0

Minimax and maximin problems for sums of translates on the real axis 实轴上平移和的极小、极大和极大问题

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-05-09 DOI: 10.1016/j.jat.2025.106190

T.M. Nikiforova

Sums of translates generalize logarithms of weighted algebraic polynomials. The paper presents the solution to the minimax and maximin problems on the real axis for sums of translates. We prove that there is a unique function that is extremal in both problems. The key in our proof is a reduction to the problem on a segment. For this, we work out an analogue of the Mhaskar–Rakhmanov–Saff theorem, too.

平移和推广了加权代数多项式的对数。本文给出了平移和在实轴上的极大极小问题的解。我们证明了在这两个问题中都存在一个唯一的极值函数。我们证明的关键是将问题简化到段上。为此，我们也提出了一个类似于Mhaskar-Rakhmanov-Saff定理的方法。

引用次数: 0

A lower bound for the Lebesgue constant of the Morrow–Patterson points 莫罗-帕特森点的勒贝格常数的下界

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-05-09 DOI: 10.1016/j.jat.2025.106191

Tomasz Beberok

The study of interpolation nodes and their associated Lebesgue constants is a cornerstone of numerical analysis, directly influencing the stability and accuracy of polynomial approximations. In this paper, we examine the Morrow–Patterson points, a specific set of interpolation nodes introduced to construct cubature formulas with the minimal number of points in a square for a fixed degree

n

. We prove that their Lebesgue constant has minimal rate of growth of at least

O (n^{2})

插值节点及其相关勒贝格常数的研究是数值分析的基石，直接影响多项式近似的稳定性和精度。在本文中，我们研究了Morrow-Patterson点，这是一组特定的插值节点，用于构造固定次数为n的方形中点数最少的立方体公式。我们证明了它们的Lebesgue常数的最小增长率至少为O（n2）。

引用次数: 0

The molecular characterizations of variable Triebel–Lizorkin spaces associated with the Hermite operator and its applications 与Hermite算子相关的变量triiebel - lizorkin空间的分子表征及其应用

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-05-06 DOI: 10.1016/j.jat.2025.106188

Qi Sun , Ciqiang Zhuo

In this article, we introduce inhomogeneous variable Triebel–Lizorkin spaces,

F_{p (\cdot), q (\cdot)}^{α (\cdot), H} (R^{n})

, associated with the Hermite operator

H ≔ - Δ + {| x |}^{2}

, where

Δ

is the Laplace operator on

R^{n}

, and mainly establish the molecular characterization of these spaces. As applications, we obtain some regularity results to fractional Hermite equations

{(- Δ + {| x |}^{2})}^{σ} u = f, {(- Δ + {| x |}^{2} + I)}^{σ} u = f,

where

σ \in (0, \infty)

, and the boundedness of spectral multiplier associated to the operator

H

on the variable Triebel–Lizorkin space

F_{p (\cdot), q (\cdot)}^{α (\cdot), H} (R^{n})

. Furthermore, we explain the relationship between

F_{p (\cdot), q (\cdot)}^{α (\cdot), H} (R^{n})

and the variable Triebel–Lizorkin spaces

F_{p (\cdot), q (\cdot)}^{α (\cdot)} (

本文引入了非齐次变量triiebel - lizorkin空间Fp(⋅),q(⋅)α(⋅)，H(Rn)，并结合Hermite算子H（Δ+|x|2），其中Δ为Rn上的拉普拉斯算子，建立了这些空间的分子表征。作为应用，我们得到了分数阶Hermite方程（−Δ+|x|2）σu=f,(−Δ+|x|2+I)σu=f的一些正则性结果，其中σ∈(0,∞)，以及变量triiebel - lizorkin空间Fp(⋅),q(⋅)α(⋅)，H（Rn）上与算子H相关的谱乘子的有界性。此外，我们通过原子分解解释了Fp（⋅）、q(⋅)α(⋅)、H（Rn）与变量triiebel - lizorkin空间Fp（⋅）、q(⋅)α(⋅)(Rn) （Diening et al.（2009）引入）之间的关系。

引用次数: 0

Orthogonal polynomials on the real line generated by the parameter sequences for a given non-single parameter positive chain sequence 对于给定的非单参数正链序列，由参数序列生成的实线上的正交多项式

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-05-06 DOI: 10.1016/j.jat.2025.106189

Daniel O. Veronese, Glalco S. Costa

In this paper, given a non-single parameter positive chain sequence

{d_{n + 1}}_{n = 1}^{\infty},

we use all the non-minimal parameter sequences for

{d_{n + 1}}_{n = 1}^{\infty}

in order to generate a whole family of sequences of orthogonal polynomials on the real line. For each non-minimal parameter sequence, the orthogonal polynomials and the associated orthogonality measure are obtained. As an application, corresponding quadratic decompositions are explicitly given. Some examples are considered in order to illustrate the results obtained.

本文给出一个非单参数正链序列{dn+1}n=1∞，利用{dn+1}n=1∞时的所有非极小参数序列，在实线上生成一组正交多项式序列。对于每一个非最小参数序列，得到了正交多项式和相应的正交测度。作为应用，明确给出了相应的二次分解。为了说明所得到的结果，考虑了一些例子。

引用次数: 0

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