Journal of Approximation Theory最新文献_第4页

Comparing the degree of constrained and unconstrained trigonometric approximation 比较有约束和无约束三角逼近的程度

IF 0.6 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-08-07 DOI: 10.1016/j.jat.2025.106220

D. Leviatan , I. Shevchuk , V. Shevchuk

Let

r, s \in N

. For a continuous

2 π

-periodic function, changing its monotonicity

2 s

times in a period, and whose degree of approximation by trigonometric polynomials of degree

< n

, is

\leq n^{- r}

,

n \geq 1

, we investigate its degree of approximation by such polynomials that, in addition, follow the changes of monotonicity. Obviously, the unconstrained degree is smaller than the constrained one, but for

r > 2 s - 2

, there is a constant

c (s, r)

such that the constrained degree is

\leq c (s, r) n^{- r}

,

n \geq 1

. On the other hand we show that, in general, this is invalid for

r \leq 2 s - 2

.

让r, s∈N。对于一个连续的2π周期函数，在一个周期内单调变化2s次，且阶为<；n的三角多项式的逼近度≤n−r， n≥1，我们研究了其单调变化的多项式的逼近度。显然，不受约束的程度小于受约束的程度，但对于r>；2s−2，存在一个常数c(s,r)，使得受约束程度≤c(s,r)n−r, n≥1。另一方面，我们证明，一般来说，这对于r≤2s−2是无效的。

引用次数: 0

Polynomial approximation in L2 with the double-sided exponential weight via complex analysis 通过复分析，用双面指数权在L2中的多项式近似

IF 0.6 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-08-06 DOI: 10.1016/j.jat.2025.106218

Pierre Bizeul , Boaz Klartag

<div><div>We study the problem of polynomial approximation in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> (<span><math><msub><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>), where <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> (<span><math><mrow><mi>d</mi><mi>x</mi></mrow></math></span>) = <span><math><mrow><mfrac><mrow><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow></msup></mrow><mrow><mn>2</mn></mrow></mfrac><mspace></mspace><mi>d</mi><mi>x</mi></mrow></math></span>. We show that for any absolutely continuous function <span><math><mi>f</mi></math></span>, <span><math><mrow><munderover><mrow><mo>∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>∞</mi></mrow></munderover><msup><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>e</mi><mo>+</mo><mi>k</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>〈</mo><mi>f</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>〉</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mo>≤</mo><mi>C</mi><mfenced><mrow><msub><mrow><mo>∫</mo></mrow><mrow><mi>R</mi></mrow></msub><msup><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>e</mi><mo>+</mo><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow><mo>)</mo></mrow><msup><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msup><mspace></mspace><mi>d</mi><msub><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mo>∫</mo></mrow><mrow><mi>R</mi></mrow></msub><msup><mrow><mrow><mo>(</mo><msup><mrow><mi>f</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mspace></mspace><mi>d</mi><msub><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfenced></mrow></math></span> for some universal constant <span><math><mrow><mi>C</mi><mo>></mo><mn>0</mn></mrow></math></span>, where <span><math><msub><mrow><mrow><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>k</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span> are the orthonormal polynomials associated with <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>. This inequality is tight in the sense that <span><math><mrow><msup><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>e</mi><mo>+</mo><mi>k</mi><mo>)</mo></mrow></mrow></math></span> on the left-hand side cannot be replaced by <span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><msup><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>e</mi><mo>+</mo><mi>k</mi><mo>)</mo></mrow></mrow></math></span> for any sequence <span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>→</mo><mi>∞</mi></mrow></math></span>. When the right-hand side is bounded, this inequality implies a logarithmic rate of approximati

研究了L2 （μ1）中的多项式逼近问题，其中μ1 (dx) = e−|x|2dx。证明了对于任意绝对连续函数f，∑k=1∞log2(e+k) < f，对于某普适常数C>；0, Pk |≤C∫Rlog2(e+|x|)f2dμ1+∫R(f ')2dμ1，其中（Pk）k∈N是与μ1相关的正交多项式。这个不等式是紧密的，因为对于任何序列ak→∞，左边的log2（e+k）不能被aklog2（e+k）所取代。当右边是有界的时候，这个不等式意味着f的一个对数逼近率，这是由Lubinsky先前得到的。我们还通过张张化论证得到了Rd上的积测度μ1⊗d的近似速率。我们的证明依赖于与权重12cosh（πx/2）相关的标准正交多项式的生成函数的显式公式，以及复分析的工具。

引用次数: 0

On positive Jacobi matrices with compact inverses 关于具有紧逆的正雅可比矩阵

IF 0.6 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-07-30 DOI: 10.1016/j.jat.2025.106217

Pavel Šťovíček , Grzegorz Świderski

We consider positive Jacobi matrices

J

with compact inverses and consequently with purely discrete spectra. A number of properties of the corresponding sequence of orthogonal polynomials is studied including the convergence of their zeros, the vague convergence of the zero counting measures and of the Christoffel–Darboux kernels on the diagonal. Particularly, if the inverse of

J

belongs to some Schatten class, we identify the asymptotic behavior of the sequence of orthogonal polynomials and express it in terms of its regularized characteristic function. In the even more special case when the inverse of

J

belongs to the trace class, we derive various formulas for the orthogonality measure, eigenvectors of

J

as well as for the functions of the second kind and related objects. These general results are given a more explicit form in the case when

- J

is a generator of a Birth–Death process. Among others, we provide a formula for the trace of the inverse of

J

. We illustrate our results by introducing and studying a modification of the

q

-Laguerre polynomials corresponding to a determinate moment problem.

我们考虑具有紧逆的正雅可比矩阵J，因而具有纯离散谱。研究了正交多项式相应序列的若干性质，包括其零点的收敛性、零计数测度的模糊收敛性和对角线上的克里斯托费尔-达布核的模糊收敛性。特别地，如果J的逆属于某个Schatten类，我们确定了正交多项式序列的渐近性质，并用正则化特征函数表示。在更特殊的情况下，当J的逆属于迹类时，我们推导了J的正交度、特征向量以及第二类函数和相关对象的各种公式。当−J是一个Birth-Death过程的生成器时，这些一般结果会有更明确的形式。其中，我们提供了j的逆轨迹的一个公式。我们通过引入和研究对应于一个定矩问题的q-Laguerre多项式的一个修正来说明我们的结果。

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引用次数: 0

Construction and approximation properties of exact neural network interpolation operators activated by entire functions 全函数激活的精确神经网络插值算子的构造及逼近性质

IF 0.6 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-07-28 DOI: 10.1016/j.jat.2025.106215

Dansheng Yu

The construction and approximation properties of exact neural network interpolation are the important and challenging topics on approximation by neural networks. Most research on exact neural network interpolation has focused on establishing existence, with very few specifically constructed interpolation neural networks proposed. The main purpose of the present paper is to provide a method for directly constructing exact neural network interpolation operators, which has the advantages that all the components in the neural network operators are explicitly known, such as the coefficients, the weights and the thresholds. By employing some important methods in approximation theory, such as the equivalence between the

K -

functional and the modulus of continuity of the function, Berens–Lorentz Lemma, and two useful estimates of the derivatives of the operators, we establish both the direct and the converse results of approximation by the new interpolation operators, and thus obtain an equivalence characterization theorem. We also introduce a type of neural network interpolation operators with four layers and a type of max-product neural network operators, rigorously analyzing their approximation properties.

精确神经网络插值的构造和逼近性质是神经网络逼近研究中的一个重要而富有挑战性的课题。大多数关于精确神经网络插值的研究都集中在建立存在性上，很少有人提出专门构造的插值神经网络。本文的主要目的是提供一种直接构造精确神经网络插值算子的方法，该方法的优点是神经网络算子中的所有成分，如系数、权值和阈值都是明确已知的。利用近似理论中的一些重要方法，如函数的K−泛函与连续模的等价性、Berens-Lorentz引理以及算子导数的两个有用的估计，我们建立了新的插值算子近似的正反结果，从而得到了等价表征定理。介绍了一类四层神经网络插值算子和一类极大积神经网络算子，并严格分析了它们的逼近性质。

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引用次数: 0

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

on

E

. If

σ

,

1 < σ < ρ (f) < \infty

, is fixed and if

m_{n} (σ)

denotes the number of interpolation points of

p_{n}

to

f

in

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

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Journal of Approximation Theory

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