Applied and Computational Harmonic Analysis最新文献_第10页

On representations of the Helmholtz Green's function 关于亥姆霍兹格林函数的表征

IF 2.5 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2024-01-24 DOI: 10.1016/j.acha.2024.101633

Gregory Beylkin

{"title":"On representations of the Helmholtz Green's function","authors":"Gregory Beylkin","doi":"10.1016/j.acha.2024.101633","DOIUrl":"10.1016/j.acha.2024.101633","url":null,"abstract":"<div>We consider the free space Helmholtz Green's function and split it into the sum of oscillatory and non-oscillatory (singular) components. The goal is to separate the impact of the singularity of the real part at the origin from the oscillatory behavior controlled by the wave number k. The oscillatory component can be chosen to have any finite number of continuous derivatives at the origin and can be applied to a function in the Fourier space in <math><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mi>d</mi></mrow></msup><mi>log</mi><mo>⁡</mo><mi>k</mi><mo>)</mo></mrow></math> operations. The non-oscillatory component has a multiresolution representation via a linear combination of Gaussians and is applied efficiently in space.Since the Helmholtz Green's function can be viewed as a point source, this partitioning can be interpreted as a splitting into propagating and evanescent components. We show that the non-oscillatory component is significant only in the vicinity of the source at distances <math><mi>O</mi><mrow><mo>(</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><msup><mrow><mi>k</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub><msup><mrow><mi>k</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msub><mrow><mi>log</mi></mrow><mrow><mn>10</mn></mrow></msub><mo>⁡</mo><mi>k</mi><mo>)</mo></mrow></math>, for some constants <math><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub></math>, <math><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub></math>, whereas the propagating component can be observed at large distances.</div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"70 ","pages":"Article 101633"},"PeriodicalIF":2.5,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139544440","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

Multivariate compactly supported C∞ functions by subdivision 通过细分实现多变量紧凑支持的 C∞ 函数

IF 2.5 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2024-01-19 DOI: 10.1016/j.acha.2024.101630

Maria Charina , Costanza Conti , Nira Dyn

{"title":"Multivariate compactly supported C∞ functions by subdivision","authors":"Maria Charina , Costanza Conti , Nira Dyn","doi":"10.1016/j.acha.2024.101630","DOIUrl":"10.1016/j.acha.2024.101630","url":null,"abstract":"<div>This paper discusses the generation of multivariate <math><msup><mrow><mi>C</mi></mrow><mrow><mo>∞</mo></mrow></msup></math> functions with compact small supports by subdivision schemes. Following the construction of such a univariate function, called Up-function, by a non-stationary scheme based on masks of spline subdivision schemes of growing degrees, we term the multivariate functions we generate Up-like functions. We generate them by non-stationary schemes based on masks of three-directional box-splines of growing supports. To analyze the convergence and smoothness of these non-stationary schemes, we develop new tools which apply to a wider class of schemes than the class we study. With our method for achieving small compact supports, we obtain in the univariate case, Up-like functions with supports <math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>+</mo><mi>ϵ</mi><mo>]</mo></math> in comparison to the support <math><mo>[</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>]</mo></math> of the Up-function. Examples of univariate and bivariate Up-like functions are given. As in the univariate case, the construction of Up-like functions can motivate the generation of <math><msup><mrow><mi>C</mi></mrow><mrow><mo>∞</mo></mrow></msup></math> compactly supported wavelets of small support in any dimension.</div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"70 ","pages":"Article 101630"},"PeriodicalIF":2.5,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1063520324000071/pdfft?md5=1ad3a9e4a30806ec403f504079a4421d&pid=1-s2.0-S1063520324000071-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139505960","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Dimension reduction, exact recovery, and error estimates for sparse reconstruction in phase space 相空间稀疏重构的降维、精确恢复和误差估计

IF 2.5 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2024-01-11 DOI: 10.1016/j.acha.2024.101631

M. Holler , A. Schlüter , B. Wirth

{"title":"Dimension reduction, exact recovery, and error estimates for sparse reconstruction in phase space","authors":"M. Holler , A. Schlüter , B. Wirth","doi":"10.1016/j.acha.2024.101631","DOIUrl":"10.1016/j.acha.2024.101631","url":null,"abstract":"<div>An important theme in modern inverse problems is the reconstruction of time-dependent data from only finitely many measurements. To obtain satisfactory reconstruction results in this setting it is essential to strongly exploit temporal consistency between the different measurement times. The strongest consistency can be achieved by reconstructing data directly in phase space, the space of positions and velocities. However, this space is usually too high-dimensional for feasible computations. We introduce a novel dimension reduction technique, based on projections of phase space onto lower-dimensional subspaces, which provably circumvents this curse of dimensionality: Indeed, in the exemplary framework of superresolution we prove that known exact reconstruction results stay true after dimension reduction, and we additionally prove new error estimates of reconstructions from noisy data in optimal transport metrics which are of the same quality as one would obtain in the non-dimension-reduced case.</div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"70 ","pages":"Article 101631"},"PeriodicalIF":2.5,"publicationDate":"2024-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1063520324000083/pdfft?md5=ecd67b0e5374d4297b6087dc7c3b9288&pid=1-s2.0-S1063520324000083-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139420410","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A divide-and-conquer algorithm for distributed optimization on networks 网络分布式优化的分而治之算法

IF 2.5 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2024-01-02 DOI: 10.1016/j.acha.2023.101623

Nazar Emirov , Guohui Song , Qiyu Sun

{"title":"A divide-and-conquer algorithm for distributed optimization on networks","authors":"Nazar Emirov , Guohui Song , Qiyu Sun","doi":"10.1016/j.acha.2023.101623","DOIUrl":"10.1016/j.acha.2023.101623","url":null,"abstract":"<div>In this paper, we consider networks with topologies described by some connected undirected graph <math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math> and with some agents (fusion centers) equipped with processing power and local peer-to-peer communication, and optimization problem <math><msub><mrow><mi>min</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>⁡</mo><mo>{</mo><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>V</mi></mrow></msub><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>}</mo></math> with local objective functions <math><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub></math> depending only on neighboring variables of the vertex <math><mi>i</mi><mo>∈</mo><mi>V</mi></math>. We introduce a divide-and-conquer algorithm to solve the above optimization problem in a distributed and decentralized manner. The proposed divide-and-conquer algorithm has exponential convergence, its computational cost is almost linear with respect to the size of the network, and it can be fully implemented at fusion centers of the network. In addition, our numerical demonstrations indicate that the proposed divide-and-conquer algorithm has superior performance than popular decentralized optimization methods in solving the least squares problem, both with and without the <math><msup><mrow><mi>ℓ</mi></mrow><mrow><mn>1</mn></mrow></msup></math> penalty, and exhibits great performance on networks equipped with asynchronous local peer-to-peer communication.</div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"70 ","pages":"Article 101623"},"PeriodicalIF":2.5,"publicationDate":"2024-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139076847","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

On the eigenvalue distribution of spatio-spectral limiting operators in higher dimensions 论高维空间谱限制算子的特征值分布

IF 2.5 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2023-12-13 DOI: 10.1016/j.acha.2023.101620

Arie Israel, Azita Mayeli

{"title":"On the eigenvalue distribution of spatio-spectral limiting operators in higher dimensions","authors":"Arie Israel, Azita Mayeli","doi":"10.1016/j.acha.2023.101620","DOIUrl":"10.1016/j.acha.2023.101620","url":null,"abstract":"<div>Prolate spheroidal wave functions are an orthogonal family of bandlimited functions on <math><mi>R</mi></math> that have the highest concentration within a specific time interval. They are also identified as the eigenfunctions of a time-frequency limiting operator (TFLO), and the associated eigenvalues belong to the interval <math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math>. Previous work has studied the asymptotic distribution and clustering behavior of the TFLO eigenvalues.In this paper, we extend these results to multiple dimensions. We prove estimates on the eigenvalues of a spatio-spectral limiting operator (SSLO) on <math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math>, which is an alternating product of projection operators associated to given spatial and frequency domains in <math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math>. If one of the domains is a hypercube, and the other domain is convex body satisfying a symmetry condition, we derive quantitative bounds on the distribution of the SSLO eigenvalues in the interval <math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math>.To prove our results, we design an orthonormal system of wave packets in <math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math> that are highly concentrated in the spatial and frequency domains. We show that these wave packets are “approximate eigenfunctions” of a spatio-spectral limiting operator. To construct the wave packets, we use a variant of the Coifman-Meyer local sine basis for <math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math>, and we lift the basis to higher dimensions using a tensor product.</div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"70 ","pages":"Article 101620"},"PeriodicalIF":2.5,"publicationDate":"2023-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138657597","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

Time-frequency analysis on flat tori and Gabor frames in finite dimensions 有限维平面环和 Gabor 框架的时频分析

IF 2.5 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2023-12-12 DOI: 10.1016/j.acha.2023.101622

L.D. Abreu , P. Balazs , N. Holighaus , F. Luef , M. Speckbacher

{"title":"Time-frequency analysis on flat tori and Gabor frames in finite dimensions","authors":"L.D. Abreu , P. Balazs , N. Holighaus , F. Luef , M. Speckbacher","doi":"10.1016/j.acha.2023.101622","DOIUrl":"10.1016/j.acha.2023.101622","url":null,"abstract":"<div>We provide the foundations of a Hilbert space theory for the short-time Fourier transform (STFT) where the flat tori <math><msubsup><mrow><mi>T</mi></mrow><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>=</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><mo>(</mo><mi>Z</mi><mo>×</mo><mi>N</mi><mi>Z</mi><mo>)</mo><mo>=</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo><mo>×</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>N</mi><mo>]</mo></math> act as phase spaces. We work on an N-dimensional subspace <math><msub><mrow><mi>S</mi></mrow><mrow><mi>N</mi></mrow></msub></math> of distributions periodic in time and frequency in the dual <math><msubsup><mrow><mi>S</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>(</mo><mi>R</mi><mo>)</mo></math> of the Feichtinger algebra <math><msub><mrow><mi>S</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math> and equip it with an inner product. To construct the Hilbert space <math><msub><mrow><mi>S</mi></mrow><mrow><mi>N</mi></mrow></msub></math> we apply a suitable double periodization operator to <math><msub><mrow><mi>S</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math>. On <math><msub><mrow><mi>S</mi></mrow><mrow><mi>N</mi></mrow></msub></math>, the STFT is applied as the usual STFT defined on <math><msubsup><mrow><mi>S</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>(</mo><mi>R</mi><mo>)</mo></math>. This STFT is a continuous extension of the finite discrete Gabor transform from the lattice onto the entire flat torus. As such, sampling theorems on flat tori lead to Gabor frames in finite dimensions. For Gaussian windows, one is lead to spaces of analytic functions and the construction allows to prove a necessary and sufficient Nyquist rate type result, which is the analogue, for Gabor frames in finite dimensions, of a well known result of Lyubarskii and Seip-Wallstén for Gabor frames with Gaussian windows and which, for N odd, produces an explicit full spark Gabor frame. The compactness of the phase space, the finite dimension of the signal spaces and our sampling theorem offer practical advantages in some applications. We illustrate this by discussing a problem of current research interest: recovering signals from the zeros of their noisy spectrograms.</div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"69 ","pages":"Article 101622"},"PeriodicalIF":2.5,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1063520323001094/pdfft?md5=a748cc66b45e71833f86016f2331a024&pid=1-s2.0-S1063520323001094-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138571556","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Laplace-Beltrami operator on the orthogonal group in ambient (Euclidean) coordinates 环境（欧几里得）坐标正交群上的拉普拉斯-贝尔特拉米算子

IF 2.5 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2023-12-12 DOI: 10.1016/j.acha.2023.101619

Petre Birtea, Ioan Caşu, Dan Comănescu

引用次数: 0

Spline manipulations for empirical mode decomposition (EMD) on bounded intervals and beyond 经验模态分解(EMD)在有界区间及以外的样条操作

IF 2.5 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2023-12-05 DOI: 10.1016/j.acha.2023.101621

Charles K. Chui , Wenjie He

引用次数: 0

Estimates on learning rates for multi-penalty distribution regression 多惩罚分布回归的学习率估计

IF 2.5 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2023-11-23 DOI: 10.1016/j.acha.2023.101609

Zhan Yu , Daniel W.C. Ho

{"title":"Estimates on learning rates for multi-penalty distribution regression","authors":"Zhan Yu , Daniel W.C. Ho","doi":"10.1016/j.acha.2023.101609","DOIUrl":"10.1016/j.acha.2023.101609","url":null,"abstract":"<div>This paper is concerned with functional learning by utilizing two-stage sampled distribution regression. We study a multi-penalty regularization algorithm for distribution regression in the framework of learning theory. The algorithm aims at regressing to real-valued outputs from probability measures. The theoretical analysis of distribution regression is far from maturity and quite challenging since only second-stage samples are observable in practical settings. In our algorithm, to transform information of distribution samples, we embed the distributions to a reproducing kernel Hilbert space <math><msub><mrow><mi>H</mi></mrow><mrow><mi>K</mi></mrow></msub></math> associated with Mercer kernel K via mean embedding technique. One of the primary contributions of this work is the introduction of a novel multi-penalty regularization algorithm, which is able to capture more potential features of distribution regression. Optimal learning rates of the algorithm are obtained under mild conditions. The work also derives learning rates for distribution regression in the hard learning scenario <math><msub><mrow><mi>f</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mo>∉</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>K</mi></mrow></msub></math>, which has not been explored in the existing literature. Moreover, we propose a new distribution-regression-based distributed learning algorithm to face large-scale data or information challenges arising from distribution data. The optimal learning rates are derived for the distributed learning algorithm. By providing new algorithms and showing their learning rates, the work improves the existing literature in various aspects.</div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"69 ","pages":"Article 101609"},"PeriodicalIF":2.5,"publicationDate":"2023-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138297364","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

Dilational symmetries of decomposition and coorbit spaces 分解与共轨道空间的扩张对称性

IF 2.5 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2023-11-17 DOI: 10.1016/j.acha.2023.101610

Hartmut Führ , Reihaneh Raisi-Tousi

引用次数: 0