Applied and Computational Harmonic Analysis最新文献

英文中文

The stability of generalized phase retrieval problem over compact groups 紧群上广义相位恢复问题的稳定性

IF 3.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2026-02-01 Epub Date: 2025-12-11 DOI: 10.1016/j.acha.2025.101838

Tal Amir , Tamir Bendory , Nadav Dym , Dan Edidin

The generalized phase retrieval problem over compact groups aims to recover a set of matrices–representing an unknown signal–from their associated Gram matrices. This framework generalizes the classical phase retrieval problem, which reconstructs a signal from the magnitudes of its Fourier transform, to a richer setting involving non-abelian compact groups. In this broader context, the unknown phases in Fourier space are replaced by unknown orthogonal matrices that arise from the action of a compact group on a finite-dimensional vector space. This problem is primarily motivated by advances in electron microscopy to determining the 3D structure of biological macromolecules from highly noisy observations. To capture realistic assumptions from machine learning and signal processing, we model the signal as belonging to one of several broad structural families: a generic linear subspace, a sparse representation in a generic basis, the output of a generic ReLU neural network, or a generic low-dimensional manifold. Our main result shows that, for a prior of sufficiently low dimension, the generalized phase retrieval problem not only admits a unique solution (up to inherent group symmetries), but also satisfies a bi-Lipschitz property. This implies robustness to both noise and model mismatch—an essential requirement for practical use, especially when measurements are severely corrupted by noise. These findings provide theoretical support for a wide class of scientific problems under modern structural assumptions, and they offer strong foundations for developing robust algorithms in high-noise regimes.

紧群上的广义相位恢复问题旨在从其相关的Gram矩阵中恢复一组表示未知信号的矩阵。该框架推广了经典的相位恢复问题，该问题从其傅里叶变换的幅度重建信号，到涉及非阿贝尔紧群的更丰富的设置。在这个更广泛的背景下，傅里叶空间中的未知相位被未知的正交矩阵所取代，这些正交矩阵是由有限维向量空间上紧群的作用产生的。这个问题的主要动机是电子显微镜技术的进步，可以从高噪声的观察中确定生物大分子的三维结构。为了从机器学习和信号处理中获取现实的假设，我们将信号建模为属于几个广泛结构族之一：一般线性子空间，一般基中的稀疏表示，一般ReLU神经网络的输出，或一般低维流形。我们的主要结果表明，对于足够低维的先验，广义相位恢复问题不仅存在唯一解（不超过固有群对称），而且满足双lipschitz性质。这意味着对噪声和模型不匹配都具有鲁棒性——这是实际使用的基本要求，特别是当测量结果受到噪声的严重破坏时。这些发现为现代结构假设下的广泛科学问题提供了理论支持，并为在高噪声条件下开发健壮的算法提供了坚实的基础。

{"title":"The stability of generalized phase retrieval problem over compact groups","authors":"Tal Amir , Tamir Bendory , Nadav Dym , Dan Edidin","doi":"10.1016/j.acha.2025.101838","DOIUrl":"10.1016/j.acha.2025.101838","url":null,"abstract":"<div><div>The generalized phase retrieval problem over compact groups aims to recover a set of matrices–representing an unknown signal–from their associated Gram matrices. This framework generalizes the classical phase retrieval problem, which reconstructs a signal from the magnitudes of its Fourier transform, to a richer setting involving non-abelian compact groups. In this broader context, the unknown phases in Fourier space are replaced by unknown orthogonal matrices that arise from the action of a compact group on a finite-dimensional vector space. This problem is primarily motivated by advances in electron microscopy to determining the 3D structure of biological macromolecules from highly noisy observations. To capture realistic assumptions from machine learning and signal processing, we model the signal as belonging to one of several broad structural families: a generic linear subspace, a sparse representation in a generic basis, the output of a generic ReLU neural network, or a generic low-dimensional manifold. Our main result shows that, for a prior of sufficiently low dimension, the generalized phase retrieval problem not only admits a unique solution (up to inherent group symmetries), but also satisfies a bi-Lipschitz property. This implies robustness to both noise and model mismatch—an essential requirement for practical use, especially when measurements are severely corrupted by noise. These findings provide theoretical support for a wide class of scientific problems under modern structural assumptions, and they offer strong foundations for developing robust algorithms in high-noise regimes.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"82 ","pages":"Article 101838"},"PeriodicalIF":3.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145732372","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Airy phase functions Airy相函数

IF 3.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2026-02-01 Epub Date: 2025-10-31 DOI: 10.1016/j.acha.2025.101820

Richard Chow, James Bremer

It is well known that phase function methods allow for the numerical solution of a large class of oscillatory second order linear ordinary differential equations in time independent of frequency. Unfortunately, these methods break down in the commonly-occurring case in which the equation has turning points. We resolve this difficulty by introducing a generalized phase function method for second order linear ordinary differential equations with turning points. More explicitly, we give an efficient numerical algorithm for computing an Airy phase function which efficiently represents the solutions of such an equation. The running time of our algorithm is independent of the magnitude of the logarithmic derivatives of the equation’s solutions, which is a measure of their rate of variation that generalizes the notion of frequency to functions which are rapidly varying but not necessarily oscillatory. Once the Airy phase function has been constructed, any reasonable initial or boundary value problem can be readily solved and, unlike step methods that only generate the values of a rapidly-varying solution at the nodes of a sparse discretization grid that is insufficient for interpolation, the output of our scheme allows for the evaluation of the solution at any point in the equation’s domain. We rigorously justify our approach by proving not only the existence of slowly-varying Airy phase functions, but also the convergence of our numerical method. Moreover, we present the results of extensive numerical experiments demonstrating the efficacy of our algorithm.

众所周知，相函数法允许对一类与时间无关的二阶振荡线性常微分方程进行数值求解。不幸的是，这些方法在方程有转折点的常见情况下失效了。通过引入一种具有拐点的二阶线性常微分方程的广义相函数方法，解决了这一难题。更明确地说，我们给出了一种计算Airy相函数的有效数值算法，该算法有效地表示了这类方程的解。我们算法的运行时间与方程解的对数导数的大小无关，这是它们变化率的度量，它将频率的概念推广到快速变化但不一定是振荡的函数。一旦构建了Airy相函数，任何合理的初始值或边值问题都可以很容易地解决，并且与只在稀疏离散网格的节点上生成快速变化的解的值的步进方法不同，这不足以进行插值，我们方案的输出允许在方程域中的任何点对解进行评估。通过证明慢变Airy相函数的存在性和数值方法的收敛性，我们严格地证明了我们的方法。此外，我们提出了大量的数值实验结果，证明了我们的算法的有效性。

{"title":"Airy phase functions","authors":"Richard Chow, James Bremer","doi":"10.1016/j.acha.2025.101820","DOIUrl":"10.1016/j.acha.2025.101820","url":null,"abstract":"<div><div>It is well known that phase function methods allow for the numerical solution of a large class of oscillatory second order linear ordinary differential equations in time independent of frequency. Unfortunately, these methods break down in the commonly-occurring case in which the equation has turning points. We resolve this difficulty by introducing a generalized phase function method for second order linear ordinary differential equations with turning points. More explicitly, we give an efficient numerical algorithm for computing an Airy phase function which efficiently represents the solutions of such an equation. The running time of our algorithm is independent of the magnitude of the logarithmic derivatives of the equation’s solutions, which is a measure of their rate of variation that generalizes the notion of frequency to functions which are rapidly varying but not necessarily oscillatory. Once the Airy phase function has been constructed, any reasonable initial or boundary value problem can be readily solved and, unlike step methods that only generate the values of a rapidly-varying solution at the nodes of a sparse discretization grid that is insufficient for interpolation, the output of our scheme allows for the evaluation of the solution at any point in the equation’s domain. We rigorously justify our approach by proving not only the existence of slowly-varying Airy phase functions, but also the convergence of our numerical method. Moreover, we present the results of extensive numerical experiments demonstrating the efficacy of our algorithm.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"81 ","pages":"Article 101820"},"PeriodicalIF":3.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145412058","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Assembly and iteration: Transition to linearity of wide neural networks 装配与迭代：广义神经网络向线性的过渡

IF 3.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2026-02-01 Epub Date: 2025-12-08 DOI: 10.1016/j.acha.2025.101834

Chaoyue Liu , Libin Zhu , Mikhail Belkin

The recently discovered remarkable property that very wide neural networks in certain regimes are linear functions of their weights has become one of the key insights into understanding the mathematical foundations of deep learning. In this work, we show that this transition to linearity of wide neural networks can be viewed as an outcome of an iterated assembly procedure employed in the construction of neural networks. From the perspective of assembly, the output of a wide network can be viewed as an assembly of a large number of similar sub-models, which will transition to linearity as their number increases. This process can be iterated multiple times to show the transition to linearity of deep networks, including general feedforward neural networks with Directed Acyclic Graph (DAG) architecture.

最近发现的一个显著特性是，在某些情况下，非常广泛的神经网络是其权重的线性函数，这已经成为理解深度学习数学基础的关键见解之一。在这项工作中，我们表明这种向广义神经网络线性的过渡可以被视为神经网络构建中采用的迭代组装过程的结果。从装配的角度来看，宽网络的输出可以看作是大量相似子模型的装配，随着子模型数量的增加，子模型会向线性过渡。这个过程可以多次迭代，以显示深度网络向线性的过渡，包括具有有向无环图（DAG）架构的一般前馈神经网络。

引用次数: 0

A max filtering local stability theorem with application to weighted phase retrieval and cryo-EM 一个最大滤波局部稳定性定理及其在加权相位恢复和低温电镜中的应用

IF 3.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2026-02-01 Epub Date: 2025-10-30 DOI: 10.1016/j.acha.2025.101821

Yousef Qaddura

Given an inner product space

V

and a group

G

of linear isometries, max filtering offers a rich class of convex

G

-invariant maps. In this paper, we identify sufficient conditions under which these maps are locally bilipschitz on

R (G)

, the set of orbits with maximal dimension, with respect to the quotient metric on the orbit space

V / G

. Central to our proof is a desingularization theorem, which applies to open, dense neighborhoods around each orbit in

R (G) / G

and may be of independent interest.

As an application, we provide guarantees for stable weighted phase retrieval. That is, we construct componentwise convex bilipschitz embeddings of weighted complex (resp. quaternionic) projective spaces. These spaces arise as quotients of direct sums of nontrivial unitary irreducible complex (resp. quaternionic) representations of the group of unit complex numbers

S^{1} ≅ SO (2)

(resp. unit quaternions

S^{3} ≅ SU (2)

We also discuss the relevance of such embeddings to a nearest-neighbor problem in single-particle cryogenic electron microscopy (cryo-EM), a leading technique for resolving the spatial structure of biological molecules.

给定一个内积空间V和一组线性等距图G，极大滤波提供了一类丰富的凸G不变映射。本文给出了这些映射相对于轨道空间V/G上的商度规在最大维数轨道集R(G)上是局部bilipschitz的充分条件。我们证明的核心是一个去广域化定理，它适用于R(G)/G中每个轨道周围的开放、密集的邻域，并且可能具有独立的兴趣。作为应用，我们为稳定的加权相位恢复提供了保证。也就是说，我们构造了加权复合体的分量凸bilipschitz嵌入。四元数)射影空间。这些空间作为非平凡酉不可约复合体的直接和的商出现。单位复数群的四元数表示S1 = SO(2)。单位四元数S3 = SU(2))。我们还讨论了这种嵌入与单粒子低温电子显微镜（cryo-EM）中最近邻问题的相关性，低温电子显微镜是解决生物分子空间结构的主要技术。

{"title":"A max filtering local stability theorem with application to weighted phase retrieval and cryo-EM","authors":"Yousef Qaddura","doi":"10.1016/j.acha.2025.101821","DOIUrl":"10.1016/j.acha.2025.101821","url":null,"abstract":"<div><div>Given an inner product space <span><math><mi>V</mi></math></span> and a group <span><math><mi>G</mi></math></span> of linear isometries, max filtering offers a rich class of convex <span><math><mi>G</mi></math></span>-invariant maps. In this paper, we identify sufficient conditions under which these maps are locally bilipschitz on <span><math><mrow><mi>R</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></math></span>, the set of orbits with maximal dimension, with respect to the quotient metric on the orbit space <span><math><mrow><mi>V</mi><mo>/</mo><mi>G</mi></mrow></math></span>. Central to our proof is a desingularization theorem, which applies to open, dense neighborhoods around each orbit in <span><math><mrow><mi>R</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>/</mo><mi>G</mi></mrow></math></span> and may be of independent interest.</div><div>As an application, we provide guarantees for stable weighted phase retrieval. That is, we construct componentwise convex bilipschitz embeddings of weighted complex (resp. quaternionic) projective spaces. These spaces arise as quotients of direct sums of nontrivial unitary irreducible complex (resp. quaternionic) representations of the group of unit complex numbers <span><math><mrow><msup><mi>S</mi><mn>1</mn></msup><mo>≅</mo><mi>SO</mi><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span> (resp. unit quaternions <span><math><mrow><msup><mi>S</mi><mn>3</mn></msup><mo>≅</mo><mi>SU</mi><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span>).</div><div>We also discuss the relevance of such embeddings to a nearest-neighbor problem in single-particle cryogenic electron microscopy (cryo-EM), a leading technique for resolving the spatial structure of biological molecules.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"81 ","pages":"Article 101821"},"PeriodicalIF":3.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145383235","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Recovering a group from few orbits 从几个轨道中恢复一个群

IF 3.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2026-02-01 Epub Date: 2025-11-29 DOI: 10.1016/j.acha.2025.101836

Dustin G. Mixon , Brantley Vose

For an unknown finite group G of automorphisms of a finite-dimensional Hilbert space, we find sharp bounds on the number of generic G-orbits needed to recover G up to group isomorphism, as well as the number needed to recover G as a concrete set of automorphisms.

对于有限维希尔伯特空间中未知的有限自同构群G，我们找到了使G恢复到群同构所需的一般G轨道的数目，以及使G恢复为一个具体的自同构集所需的数目的明显界限。

引用次数: 0

Signal reconstruction using determinantal sampling 使用确定性采样的信号重建

IF 3.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2026-02-01 Epub Date: 2025-11-03 DOI: 10.1016/j.acha.2025.101822

Ayoub Belhadji , Rémi Bardenet , Pierre Chainais

We study the approximation of a square-integrable function from a finite number of evaluations on a random set of nodes according to a well-chosen distribution. This is particularly relevant when the function is assumed to belong to a reproducing kernel Hilbert space (RKHS). This work proposes to combine several natural finite-dimensional approximations based on two possible probability distributions of nodes. These distributions are related to determinantal point processes, and use the kernel of the RKHS to favor RKHS-adapted regularity in the random design. While previous work on determinantal sampling relied on the RKHS norm, we prove mean-square guarantees in L² norm. We show that determinantal point processes and mixtures thereof can yield fast convergence rates. Our results also shed light on how the rate changes as more smoothness is assumed, a phenomenon known as superconvergence. Besides, determinantal sampling improves on i.i.d. sampling from the Christoffel function which is standard in the literature. More importantly, determinantal sampling guarantees the so-called instance optimality property for a smaller number of function evaluations than i.i.d. sampling.

我们研究了一个平方可积函数的逼近，它是根据一个选定的分布，由有限个随机节点集上的求值得到的。当假设函数属于再现核希尔伯特空间（RKHS）时，这一点尤为重要。这项工作提出结合基于两个可能的节点概率分布的几种自然有限维近似。这些分布与确定性点过程有关，并且在随机设计中使用RKHS的核来支持RKHS适应的规律性。虽然先前的确定性抽样工作依赖于RKHS范数，但我们证明了L2范数的均方保证。我们证明了行列式点过程及其混合过程可以产生快速的收敛速度。我们的研究结果还揭示了当假设更平滑时速率是如何变化的，这种现象被称为超收敛。此外，确定性抽样改进了文献中标准的Christoffel函数的i.i.d抽样。更重要的是，决定论抽样保证了所谓的实例最优性，与i.i.d抽样相比，函数求值的次数更少。

{"title":"Signal reconstruction using determinantal sampling","authors":"Ayoub Belhadji , Rémi Bardenet , Pierre Chainais","doi":"10.1016/j.acha.2025.101822","DOIUrl":"10.1016/j.acha.2025.101822","url":null,"abstract":"<div><div>We study the approximation of a square-integrable function from a finite number of evaluations on a random set of nodes according to a well-chosen distribution. This is particularly relevant when the function is assumed to belong to a reproducing kernel Hilbert space (RKHS). This work proposes to combine several natural finite-dimensional approximations based on two possible probability distributions of nodes. These distributions are related to determinantal point processes, and use the kernel of the RKHS to favor RKHS-adapted regularity in the random design. While previous work on determinantal sampling relied on the RKHS norm, we prove mean-square guarantees in <em>L</em><sup>2</sup> norm. We show that determinantal point processes and mixtures thereof can yield fast convergence rates. Our results also shed light on how the rate changes as more smoothness is assumed, a phenomenon known as superconvergence. Besides, determinantal sampling improves on i.i.d. sampling from the Christoffel function which is standard in the literature. More importantly, determinantal sampling guarantees the so-called instance optimality property for a smaller number of function evaluations than i.i.d. sampling.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"82 ","pages":"Article 101822"},"PeriodicalIF":3.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145434666","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Direct interpolative construction of the discrete Fourier transform as a matrix product operator 离散傅里叶变换作为矩阵乘积算子的直接插值构造

IF 3.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2026-02-01 Epub Date: 2025-10-26 DOI: 10.1016/j.acha.2025.101817

Jielun Chen , Michael Lindsey

The quantum Fourier transform (QFT), which can be viewed as a reindexing of the discrete Fourier transform (DFT), has been shown to be compressible as a low-rank matrix product operator (MPO) or quantized tensor train (QTT) operator [1]. However, the original proof of this fact does not furnish a construction of the MPO with a guaranteed error bound. Meanwhile, the existing practical construction of this MPO, based on the compression of a quantum circuit, is not as efficient as possible. We present a simple closed-form construction of the QFT MPO using the interpolative decomposition, with guaranteed near-optimal compression error for a given rank. This construction can speed up the application of the QFT and the DFT, respectively, in quantum circuit simulations and QTT applications. We also connect our interpolative construction to the approximate quantum Fourier transform (AQFT) by demonstrating that the AQFT can be viewed as an MPO constructed using a different interpolation scheme.

量子傅里叶变换（QFT）可以看作是离散傅里叶变换（DFT）的重新索引，它被证明是可压缩的低秩矩阵积算子（MPO）或量化张量序列算子（QTT）[1]。然而，对这一事实的原始证明并没有提供一个具有保证误差界的MPO构造。同时，现有的基于量子电路压缩的MPO的实际结构并没有达到尽可能高的效率。我们提出了一个使用插值分解的QFT MPO的简单封闭形式构造，保证了给定秩的近最优压缩误差。这种结构可以加快QFT和DFT分别在量子电路模拟和QTT应用中的应用。我们还通过证明AQFT可以被视为使用不同插值方案构建的MPO，将我们的插值构造与近似量子傅里叶变换（AQFT）联系起来。

引用次数: 0

Ramanujan sums in signal recovery and uncertainty principle inequalities 信号恢复和不确定性原理不等式中的Ramanujan和

IF 3.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2026-02-01 Epub Date: 2025-12-25 DOI: 10.1016/j.acha.2025.101852

Sahil Kalra, Niraj K. Shukla

This paper explores the perfect reconstruction property of filter banks based on Ramanujan sums and their applications in signal recovery. Originally introduced by Srinivasa Ramanujan, Ramanujan sums serve as powerful tools for extracting periodic components from signals and form the foundation of Ramanujan filter banks. We investigate the perfect reconstruction property of these filter banks and analyze their robustness against erasures for discrete-time signals in the finite-dimensional space

ℓ^{2} (Z_{N})

(i.e.,

C^{N}

). The study is further extended to non-uniform Ramanujan filter banks, showcasing their ability to address the limitations of uniform ones. Employing the reconstruction properties of uniform Ramanujan filter banks, we present an uncertainty principle associated with a tight frame generated by shifts of Ramanujan sums. This principle establishes representation inequalities in terms of Euler’s totient function ϕ(n), which provide sufficient conditions for the perfect recovery of signals in scenarios where signal information is lost during transmission or corrupted by noise. Finally, we illustrate that utilizing the signal’s periodicity information through Ramanujan filter banks significantly improves the efficiency of signal recovery optimization algorithms, resulting in enhanced signal-to-noise ratio (SNR) gains and more precise reconstruction.

本文探讨了基于拉马努金和的滤波器组的完美重构特性及其在信号恢复中的应用。最初由Srinivasa Ramanujan引入，Ramanujan和作为从信号中提取周期分量的强大工具，并构成了Ramanujan滤波器组的基础。我们研究了这些滤波器组的完美重构特性，并分析了它们在有限维空间l2 (ZN)（即CN）中对离散时间信号的抗擦除的鲁棒性。该研究进一步扩展到非均匀拉马努金滤波器组，展示了它们解决均匀滤波器组局限性的能力。利用均匀拉马努金滤波器组的重构性质，给出了拉马努金和移位产生的紧框架的不确定性原理。该原理建立了欧拉完备函数φ (n)的表示不等式，为信号信息在传输过程中丢失或被噪声破坏的情况下信号的完美恢复提供了充分条件。最后，我们证明了通过拉马努金滤波器组利用信号的周期性信息可以显著提高信号恢复优化算法的效率，从而提高信噪比（SNR）增益和更精确的重建。

{"title":"Ramanujan sums in signal recovery and uncertainty principle inequalities","authors":"Sahil Kalra, Niraj K. Shukla","doi":"10.1016/j.acha.2025.101852","DOIUrl":"10.1016/j.acha.2025.101852","url":null,"abstract":"<div><div>This paper explores the perfect reconstruction property of filter banks based on Ramanujan sums and their applications in signal recovery. Originally introduced by Srinivasa Ramanujan, Ramanujan sums serve as powerful tools for extracting periodic components from signals and form the foundation of Ramanujan filter banks. We investigate the perfect reconstruction property of these filter banks and analyze their robustness against erasures for discrete-time signals in the finite-dimensional space <span><math><mrow><msup><mi>ℓ</mi><mn>2</mn></msup><mrow><mo>(</mo><msub><mi>Z</mi><mi>N</mi></msub><mo>)</mo></mrow></mrow></math></span> (i.e., <span><math><msup><mi>C</mi><mi>N</mi></msup></math></span>). The study is further extended to non-uniform Ramanujan filter banks, showcasing their ability to address the limitations of uniform ones. Employing the reconstruction properties of uniform Ramanujan filter banks, we present an uncertainty principle associated with a tight frame generated by shifts of Ramanujan sums. This principle establishes representation inequalities in terms of Euler’s totient function <em>ϕ</em>(<em>n</em>), which provide sufficient conditions for the perfect recovery of signals in scenarios where signal information is lost during transmission or corrupted by noise. Finally, we illustrate that utilizing the signal’s periodicity information through Ramanujan filter banks significantly improves the efficiency of signal recovery optimization algorithms, resulting in enhanced signal-to-noise ratio (SNR) gains and more precise reconstruction.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"82 ","pages":"Article 101852"},"PeriodicalIF":3.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145845085","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

High-order synchrosqueezed chirplet transforms for multicomponent signal analysis 用于多分量信号分析的高阶同步压缩啁啾变换

IF 3.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2026-02-01 Epub Date: 2025-12-01 DOI: 10.1016/j.acha.2025.101839

Yi-Ju Yen , De-Yan Lu , Sing-Yuan Yeh , Jian-Jiun Ding , Chun-Yen Shen

This study focuses on the analysis of signals containing multiple components with crossover instantaneous frequencies (IF). This problem was initially solved with the chirplet transform (CT). Also, it can be sharpened by adding the synchrosqueezing step, which is called the synchrosqueezed chirplet transform (SCT). However, we found that the SCT goes wrong with the high chirp modulation signal due to the wrong estimation of the IF. In this paper, we present the improvement of the post-transformation of the CT. The main goal of this paper is to amend the estimation introduced in the SCT and carry out the high-order synchrosqueezed chirplet transform. The proposed method reduces the wrong estimation when facing a stronger variety of chirp-modulated multi-component signals. The theoretical analysis of the new reassignment ingredient is provided. Numerical experiments on some synthetic signals are presented to verify the effectiveness of the proposed high-order SCT.

本研究主要针对具有交叉瞬时频率（IF）的多分量信号进行分析。这个问题最初是用小波变换（CT）来解决的。此外，它可以通过增加同步压缩步骤来锐化，这被称为同步压缩啁啾变换（SCT）。然而，我们发现由于对中频的错误估计，SCT在高啁啾调制信号下会出错。在本文中，我们提出了改进后变换的CT。本文的主要目的是对SCT中引入的估计进行修正，并进行高阶同步压缩小波变换。该方法减少了在面对多种啁啾调制多分量信号时的错误估计。对这种新的重分配成分进行了理论分析。在一些合成信号上进行了数值实验，验证了该方法的有效性。

引用次数: 0

Adaptive multipliers for extrapolation in frequency 频率外推的自适应乘法器

IF 3.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2026-01-01 Epub Date: 2025-09-15 DOI: 10.1016/j.acha.2025.101815

Diego Castelli Lacunza , Carlos A. Sing Long

Resolving the details of an object from coarse-scale measurements is a classical problem in applied mathematics. This problem is usually formulated as extrapolating the Fourier transform of the object from a bounded region to the entire space, that is, in terms of performing extrapolation in frequency. This problem is ill-posed unless one assumes that the object has some additional structure. When the object is compactly supported, then it is well-known that its Fourier transform can be extended to the entire space. However, it is also well-known that this problem is severely ill-conditioned.

In this work, we assume that the object is known to belong to a collection of compactly supported functions and, instead of performing extrapolation in frequency to the entire space, we study the problem of extrapolating to a larger bounded set using dilations in frequency and a single Fourier multiplier. This is reminiscent of the refinement equation in multiresolution analysis. Under suitable conditions, we prove the existence of a worst-case optimal multiplier over the entire collection, and we show that all such multipliers share the same canonical structure. When the collection is finite, we show that any worst-case optimal multiplier can be represented in terms of an Hermitian matrix. This allows us to introduce a fixed-point iteration to find the optimal multiplier. This leads us to introduce a family of multipliers, which we call

Σ

-multipliers, that can be used to perform extrapolation in frequency. We establish connections between

Σ

-multipliers and multiresolution analysis. We conclude with some numerical experiments illustrating the practical consequences of our results.

从粗尺度测量中求解物体的细节是应用数学中的一个经典问题。这个问题通常被表述为将物体的傅里叶变换从有界区域外推到整个空间，也就是说，在频率上进行外推。这个问题是不适定的，除非假设对象有一些附加结构。当目标被紧支撑时，其傅里叶变换可以扩展到整个空间。然而，众所周知，这个问题是严重病态的。在这项工作中，我们假设已知对象属于紧支持函数的集合，而不是对整个空间进行频率外推，我们研究了使用频率膨胀和单个傅里叶乘数向更大的有界集合外推的问题。这让人想起多分辨率分析中的细化方程。在适当的条件下，我们证明了在整个集合上存在一个最坏情况最优乘子，并证明了所有这些乘子具有相同的规范结构。当集合有限时，我们证明了任何最坏情况下的最优乘子都可以用厄米矩阵表示。这允许我们引入一个定点迭代来找到最优乘数。这导致我们引入一系列乘法器，我们称之为Σ-multipliers，它可以用来执行频率外推。我们建立了Σ-multipliers和多分辨率分析之间的联系。最后，我们用一些数值实验来说明我们的结果的实际意义。

{"title":"Adaptive multipliers for extrapolation in frequency","authors":"Diego Castelli Lacunza , Carlos A. Sing Long","doi":"10.1016/j.acha.2025.101815","DOIUrl":"10.1016/j.acha.2025.101815","url":null,"abstract":"<div><div>Resolving the details of an object from coarse-scale measurements is a classical problem in applied mathematics. This problem is usually formulated as extrapolating the Fourier transform of the object from a bounded region to the entire space, that is, in terms of performing <em>extrapolation in frequency</em>. This problem is ill-posed unless one assumes that the object has some additional structure. When the object is compactly supported, then it is well-known that its Fourier transform can be extended to the entire space. However, it is also well-known that this problem is severely ill-conditioned.</div><div>In this work, we assume that the object is known to belong to a collection of compactly supported functions and, instead of performing extrapolation in frequency to the entire space, we study the problem of extrapolating to a larger bounded set using dilations in frequency and a single Fourier multiplier. This is reminiscent of the refinement equation in multiresolution analysis. Under suitable conditions, we prove the existence of a worst-case optimal multiplier over the entire collection, and we show that all such multipliers share the same canonical structure. When the collection is finite, we show that any worst-case optimal multiplier can be represented in terms of an Hermitian matrix. This allows us to introduce a fixed-point iteration to find the optimal multiplier. This leads us to introduce a family of multipliers, which we call <span><math><mstyle><mi>Σ</mi></mstyle></math></span>-multipliers, that can be used to perform extrapolation in frequency. We establish connections between <span><math><mstyle><mi>Σ</mi></mstyle></math></span>-multipliers and multiresolution analysis. We conclude with some numerical experiments illustrating the practical consequences of our results.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"80 ","pages":"Article 101815"},"PeriodicalIF":3.2,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145121213","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

首页上一页

下一页尾页

类型

全部化学•材料生命科学医学物理工程技术环境•农林材料科学地球科学法学管理学化学环境科学与生态学计算机科学教育学经济学农林科学人文科学生物学数学物理与天体物理心理学综合性期刊其他工业工程理学历史学农学文学信息工程

数据库

全部 ACS Publications Elsevier ieeexplore Springer The Royal Society of Chemistry Wiley

期刊

Applied and Computational Harmonic Analysis

全部 Acc. Chem. Res. ACS Applied Bio Materials ACS Appl. Electron. Mater. ACS Appl. Energy Mater. ACS Appl. Mater. Interfaces ACS Appl. Nano Mater. ACS Appl. Polym. Mater. ACS BIOMATER-SCI ENG ACS Catal. ACS Cent. Sci. ACS Chem. Biol. ACS Chemical Health & Safety ACS Chem. Neurosci. ACS Comb. Sci. ACS Earth Space Chem. ACS Energy Lett. ACS Infect. Dis. ACS Macro Lett. ACS Mater. Lett. ACS Med. Chem. Lett. ACS Nano ACS Omega ACS Photonics ACS Sens. ACS Sustainable Chem. Eng. ACS Synth. Biol. Anal. Chem. BIOCHEMISTRY-US Bioconjugate Chem. BIOMACROMOLECULES Chem. Res. Toxicol. Chem. Rev. Chem. Mater. CRYST GROWTH DES ENERG FUEL Environ. Sci. Technol. Environ. Sci. Technol. Lett. Eur. J. Inorg. Chem. IND ENG CHEM RES Inorg. Chem. J. Agric. Food. Chem. J. Chem. Eng. Data J. Chem. Educ. J. Chem. Inf. Model. J. Chem. Theory Comput. J. Med. Chem. J. Nat. Prod. J PROTEOME RES J. Am. Chem. Soc. LANGMUIR MACROMOLECULES Mol. Pharmaceutics Nano Lett. Org. Lett. ORG PROCESS RES DEV ORGANOMETALLICS J. Org. Chem. J. Phys. Chem. J. Phys. Chem. A J. Phys. Chem. B J. Phys. Chem. C J. Phys. Chem. Lett. Analyst Anal. Methods Biomater. Sci. Catal. Sci. Technol. Chem. Commun. Chem. Soc. Rev. CHEM EDUC RES PRACT CRYSTENGCOMM Dalton Trans. Energy Environ. Sci. ENVIRON SCI-NANO ENVIRON SCI-PROC IMP ENVIRON SCI-WAT RES Faraday Discuss. Food Funct. Green Chem. Inorg. Chem. Front. Integr. Biol. J. Anal. At. Spectrom. J. Mater. Chem. A J. Mater. Chem. B J. Mater. Chem. C Lab Chip Mater. Chem. Front. Mater. Horiz. MEDCHEMCOMM Metallomics Mol. Biosyst. Mol. Syst. Des. Eng. Nanoscale Nanoscale Horiz. Nat. Prod. Rep. New J. Chem. Org. Biomol. Chem. Org. Chem. Front. PHOTOCH PHOTOBIO SCI PCCP Polym. Chem.

﹀