Journal of Approximation Theory最新文献_第5页

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

Global rational approximations of functions with factorially divergent asymptotic series 具有阶乘发散渐近级数的函数的全局有理逼近

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-04-23 DOI: 10.1016/j.jat.2025.106178

N. Castillo, O. Costin, R.D. Costin

We construct a new type of convergent, and asymptotic, representations, dyadic expansions. Their convergence is geometric and the region of convergence often extends from infinity down to

0^{+}

. We show that dyadic expansions are numerically efficient representations.

For special functions such as Bessel, Airy, Ei, erfc, Gamma, etc. the region of convergence of dyadic series is the complex plane minus a ray, with this cut chosen at will. Dyadic expansions thus provide uniform, geometrically convergent asymptotic expansions including near antistokes rays.

We prove that relatively general functions, Écalle resurgent ones, possess convergent dyadic expansions.

These expansions extend to operators, resulting in representations of the resolvent of self-adjoint operators as series in terms of the associated unitary evolution operator evaluated at some prescribed discrete times (alternatively, for positive operators, in terms of the generated semigroup).

我们构造了一种新的收敛的，渐近的，表示，二进展开式。它们的收敛是几何的，收敛的区域通常从无穷远处延伸到0+。我们证明了二进展开式是数值上有效的表示。对于特殊的函数，如Bessel， Airy, Ei, erfc， Gamma等，并矢级数的收敛区域是复平面减去一条射线，这个切割可以随意选择。因此，并矢展开式提供了均匀的、几何收敛的渐近展开式，包括近反斯托克斯射线。我们证明了相对一般的函数Écalle复活函数具有收敛的二进展开式。这些展开式扩展到算子，导致自伴随算子的解表示为在某些规定的离散时间内计算的相关的酉演化算子的级数（或者，对于正算子，根据生成的半群）。

引用次数: 0

Approximating positive homogeneous functions with scale invariant neural networks 用尺度不变神经网络逼近正齐次函数

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-04-23 DOI: 10.1016/j.jat.2025.106177

Stefan Bamberger , Reinhard Heckel , Felix Krahmer

We investigate the approximation of positive homogeneous functions, i.e., functions

f

satisfying

f (λ x) = λ f (x)

for all

λ \geq 0

, with neural networks. Extending previous work, we establish new results explaining under which conditions such functions can be approximated with neural networks. As a key application for this, we analyze to what extent it is possible to solve linear inverse problems with

ReLu

networks. Due to the scaling invariance arising from the linearity, an optimal reconstruction function for such a problem is positive homogeneous. In a

ReLu

network, this condition translates to considering networks without bias terms. For the recovery of sparse vectors from few linear measurements, our results imply that

ReLu

networks with two hidden layers allow approximate recovery with arbitrary precision and arbitrary sparsity level

s

in a stable way. In contrast, we also show that with only one hidden layer such networks cannot even recover 1-sparse vectors, not even approximately, and regardless of the width of the network. These findings even apply to a wider class of recovery problems including low-rank matrix recovery and phase retrieval. Our results also shed some light on the seeming contradiction between previous works showing that neural networks for inverse problems typically have very large Lipschitz constants, but still perform very well also for adversarial noise. Namely, the error bounds in our expressivity results include a combination of a small constant term and a term that is linear in the noise level, indicating that robustness issues may occur only for very small noise levels.

利用神经网络研究了正齐次函数的逼近，即对于所有λ≥0，函数f满足f(λx)=λf(x)。扩展先前的工作，我们建立了新的结果，解释了在哪些条件下这些函数可以用神经网络近似。作为该方法的一个关键应用，我们分析了ReLu网络在多大程度上可以解决线性逆问题。由于线性引起的标度不变性，该问题的最优重构函数是正齐次的。在ReLu网络中，这个条件转化为考虑没有偏置项的网络。对于从少量线性测量中恢复稀疏向量，我们的结果表明，具有两个隐藏层的ReLu网络可以以稳定的方式以任意精度和任意稀疏度水平s近似恢复。相反，我们还表明，只有一个隐藏层，这样的网络甚至不能恢复1-稀疏向量，甚至不能近似地恢复，并且与网络的宽度无关。这些发现甚至适用于更广泛的恢复问题，包括低秩矩阵恢复和相位恢复。我们的结果还揭示了之前的研究之间的矛盾，表明反问题的神经网络通常具有非常大的Lipschitz常数，但对于对抗噪声仍然表现得很好。也就是说，我们的表达式结果中的误差界限包括一个小的常数项和一个在噪声水平上是线性的项的组合，这表明鲁棒性问题可能只发生在非常小的噪声水平上。

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引用次数: 0

On a lemma by Brézis and Haraux 关于brsamzis和Haraux的引理

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-04-18 DOI: 10.1016/j.jat.2025.106175

Minh N. Bùi

We propose several applications of an often overlooked part of the 1976 paper by Brézis and Haraux, in which the Brézis–Haraux theorem was established. Our results unify and extend various existing ones on the range of a linearly composite monotone operator and provide new insight into their seminal paper.

我们提出了一些关于brsamzis和Haraux在1976年的论文中经常被忽视的部分的应用，在该论文中建立了brsamzis - Haraux定理。我们的结果统一和推广了现有的关于线性复合单调算子范围的各种结果，并为他们的开创性论文提供了新的见解。

引用次数: 0

Nyström subsampling for functional linear regression Nyström函数线性回归的子抽样

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-04-16 DOI: 10.1016/j.jat.2025.106176

Jun Fan , Jiading Liu , Lei Shi

Kernel methods have proven to be highly effective for functional data analysis, demonstrating significant theoretical and practical success over the past two decades. However, their computational complexity and storage requirements hinder their direct application to large-scale functional data learning problems. In this paper, we address this limitation by investigating the theoretical properties of the Nyström subsampling method within the framework of the functional linear regression model and reproducing kernel Hilbert space. Our proposed algorithm not only overcomes the computational challenges but also achieves the minimax optimal rate of convergence for the excess prediction risk, provided an appropriate subsampling size is chosen. Our error analysis relies on the approximation of integral operators induced by the reproducing kernel and covariance function.

核方法已被证明对功能数据分析非常有效，在过去二十年中取得了重大的理论和实践成功。然而，它们的计算复杂性和存储要求阻碍了它们直接应用于大规模功能数据学习问题。在本文中，我们通过研究Nyström子抽样方法在函数线性回归模型框架内的理论性质和再现核希尔伯特空间来解决这一限制。我们提出的算法不仅克服了计算上的挑战，而且在选择适当的子样本大小的情况下，可以实现对超额预测风险的最小最大最优收敛速度。我们的误差分析依赖于由再现核和协方差函数引起的积分算子的近似。

引用次数: 0

Extension of the best polynomial operator in generalized Orlicz Spaces 广义Orlicz空间中最佳多项式算子的推广

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-03-26 DOI: 10.1016/j.jat.2025.106174

Sonia Acinas , Sergio Favier , Rosa Lorenzo

In this paper, we consider the best multivalued polynomial approximation operator for functions in an Orlicz Space

L^{φ} (Ω)

. We obtain its characterization involving

ψ^{-}

and

ψ^{+}

, which are the left and right derivative functions of

φ

. And then, we extend the operator to

L^{ψ^{+}} (Ω)

. We also get pointwise convergence of this extension, where the Calderón–Zygmund class

t_{m}^{p} (x)

adapted to

L^{ψ^{+}} (Ω)

plays an important role.

本文考虑Orlicz空间Lφ（Ω）中函数的最佳多值多项式逼近算子。我们得到了φ的左导数函数ψ−和右导数函数ψ+的表征。然后，我们把算子扩展到Lψ+（Ω）我们也得到了这个扩展的点向收敛，其中Calderón-Zygmund类tmp(x)适应于Lψ+（Ω）起着重要的作用。

引用次数: 0

Rearrangement-invariant norm inequalities for convolution operators 卷积算子的重排不变范数不等式

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-03-26 DOI: 10.1016/j.jat.2025.106173

Ron Kerman , S. Spektor

Let

k \in (L_{1} + L_{\infty}) (R^{n})

, where, as usual,

L_{1} (R^{n})

denotes the class of Lebesgue-integrable functions on

R^{n}

and

L_{\infty} (R^{n})

denotes the class of functions on

R^{n}

that are Lebesgue-measurable and bounded almost everywhere. Given

f \in (L_{1} \cap L_{\infty}) (R^{n})

, set

(T_{k} f) (x) = \int_{R^{n}} k (x - y) f (y) d y, x \in R^{n} .

We study inequalities of the form

ρ (T_{k} f) \leq C σ (f),

in which

C > 0

is independent of

f \in (L_{1} \cap L_{\infty}) (R^{n})

. The functionals

ρ

and

σ

are so-called rearrangement-invariant (r.i.) norms on

M_{+} (R^{n})

设k∈(L1+L∞)(Rn)，其中，通常，L1（Rn）表示Rn上的勒贝格可积函数类，L∞(Rn)表示Rn上的几乎处处勒贝格可测且有界的函数类。鉴于f∈(L1∩L∞)(Rn)组(Tkf) (x) =∫Rnk (x−y) f dy (y), x∈Rn。我们研究了ρ(Tkf)≤Cσ(f)的不等式，其中C>；0与f∈(L1∩L∞)（Rn）无关。泛函ρ和σ是所谓的M+(Rn)上的重排不变（r.i）范数，这是Rn上的一类非负可测函数。在r.i.范数的一般情况下首先证明的结果在Orlicz范数的特殊情况下都是专门化和扩展的。

引用次数: 0

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