Applied and Computational Harmonic Analysis最新文献

{"title":"Short-time Fourier transform and superoscillations","authors":"Daniel Alpay ,&nbsp;Antonino De Martino ,&nbsp;Kamal Diki ,&nbsp;Daniele C. Struppa","doi":"10.1016/j.acha.2024.101689","DOIUrl":"10.1016/j.acha.2024.101689","url":null,"abstract":"<div><p>In this paper we investigate new results on the theory of superoscillations using time-frequency analysis tools and techniques such as the short-time Fourier transform (STFT) and the Zak transform. We start by studying how the short-time Fourier transform acts on superoscillation sequences. We then apply the supershift property to prove that the short-time Fourier transform preserves the superoscillatory behavior by taking the limit. It turns out that these computations lead to interesting connections with various features of time-frequency analysis such as Gabor spaces, Gabor kernels, Gabor frames, 2D-complex Hermite polynomials, and polyanalytic functions. We treat different cases depending on the choice of the window function moving from the general case to more specific cases involving the Gaussian and the Hermite windows. We consider also an evolution problem with an initial datum given by superoscillation multiplied by the time-frequency shifts of a generic window function. Finally, we compute the action of STFT on the approximating sequences with a given Hermite window.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"73 ","pages":"Article 101689"},"PeriodicalIF":2.6,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1063520324000666/pdfft?md5=96fd7a993bcf9429a7aae8f923cc37d9&pid=1-s2.0-S1063520324000666-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141841959","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}

{"title":"Local structure and effective dimensionality of time series data sets","authors":"Monika Dörfler,&nbsp;Franz Luef,&nbsp;Eirik Skrettingland","doi":"10.1016/j.acha.2024.101692","DOIUrl":"10.1016/j.acha.2024.101692","url":null,"abstract":"<div><p>The goal of this paper is to develop novel tools for understanding the local structure of systems of functions, e.g. time-series data points. The proposed tools include a total correlation function, the Cohen class of the data set, the data operator and the average lack of concentration. The Cohen class of the data operator gives a time-frequency representation of the data set. Furthermore, we show that the von Neumann entropy of the data operator captures local features of the data set and that it is related to the notion of effective dimensionality. The accumulated Cohen class of the data operator gives us a low-dimensional representation of the data set and we quantify this in terms of the average lack of concentration and the von Neumann entropy of the data operator and an improvement of the Berezin-Lieb inequality using the projection functional of the data augmentation operator. The framework for our approach is provided by quantum harmonic analysis.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"73 ","pages":"Article 101692"},"PeriodicalIF":2.6,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1063520324000691/pdfft?md5=44281e1bf43209bd5a937cf474a8b490&pid=1-s2.0-S1063520324000691-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141891937","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}

{"title":"Embeddings between Barron spaces with higher-order activation functions","authors":"Tjeerd Jan Heeringa ,&nbsp;Len Spek ,&nbsp;Felix L. Schwenninger ,&nbsp;Christoph Brune","doi":"10.1016/j.acha.2024.101691","DOIUrl":"10.1016/j.acha.2024.101691","url":null,"abstract":"<div><p>The approximation properties of infinitely wide shallow neural networks heavily depend on the choice of the activation function. To understand this influence, we study embeddings between Barron spaces with different activation functions. These embeddings are proven by providing push-forward maps on the measures <em>μ</em> used to represent functions <em>f</em>. An activation function of particular interest is the rectified power unit (RePU) given by <span><math><msub><mrow><mi>RePU</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>max</mi><mo>⁡</mo><msup><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msup></math></span>. For many commonly used activation functions, the well-known Taylor remainder theorem can be used to construct a push-forward map, which allows us to prove the embedding of the associated Barron space into a Barron space with a RePU as activation function. Moreover, the Barron spaces associated with the <span><math><msub><mrow><mi>RePU</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span> have a hierarchical structure similar to the Sobolev spaces <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span>.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"73 ","pages":"Article 101691"},"PeriodicalIF":2.6,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S106352032400068X/pdfft?md5=d7b9ed667155aca5daca913402281afb&pid=1-s2.0-S106352032400068X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141992699","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}

{"title":"EDMD for expanding circle maps and their complex perturbations","authors":"Oscar F. Bandtlow ,&nbsp;Wolfram Just ,&nbsp;Julia Slipantschuk","doi":"10.1016/j.acha.2024.101690","DOIUrl":"10.1016/j.acha.2024.101690","url":null,"abstract":"<div><p>We show that spectral data of the Koopman operator arising from an analytic expanding circle map <em>τ</em> can be effectively calculated using an EDMD-type algorithm combining a collocation method of order <em>m</em> with a Galerkin method of order <em>n</em>. The main result is that if <span><math><mi>m</mi><mo>≥</mo><mi>δ</mi><mi>n</mi></math></span>, where <em>δ</em> is an explicitly given positive number quantifying by how much <em>τ</em> expands concentric annuli containing the unit circle, then the method converges and approximates the spectrum of the Koopman operator, taken to be acting on a space of analytic hyperfunctions, exponentially fast in <em>n</em>. Additionally, these results extend to more general expansive maps on suitable annuli containing the unit circle.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"73 ","pages":"Article 101690"},"PeriodicalIF":2.6,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1063520324000678/pdfft?md5=19289db18c6f007b7c00dd403231dfce&pid=1-s2.0-S1063520324000678-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141891939","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}

{"title":"Matrix recovery from permutations","authors":"Manolis C. Tsakiris","doi":"10.1016/j.acha.2024.101688","DOIUrl":"10.1016/j.acha.2024.101688","url":null,"abstract":"<div><p>In data science, a number of applications have been emerging involving data recovery from permutations. Here, we study this problem theoretically for data organized in a rank-deficient matrix. Specifically, we give unique recovery guarantees for matrices of bounded rank that have undergone arbitrary permutations of their entries. We use methods and results of commutative algebra and algebraic geometry, for which we include a preparation for a general audience.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"73 ","pages":"Article 101688"},"PeriodicalIF":2.6,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141711977","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}

{"title":"Computing sparse Fourier sum of squares on finite abelian groups in quasi-linear time","authors":"Jianting Yang ,&nbsp;Ke Ye ,&nbsp;Lihong Zhi","doi":"10.1016/j.acha.2024.101686","DOIUrl":"10.1016/j.acha.2024.101686","url":null,"abstract":"<div><p>The problem of verifying the nonnegativity of a function on a finite abelian group is a long-standing challenging problem. The basic representation theory of finite groups indicates that a function <em>f</em> on a finite abelian group <em>G</em> can be written as a linear combination of characters of irreducible representations of <em>G</em> by <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>χ</mi><mo>∈</mo><mover><mrow><mi>G</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow></msub><mover><mrow><mi>f</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>(</mo><mi>χ</mi><mo>)</mo><mi>χ</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>, where <span><math><mover><mrow><mi>G</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span> is the dual group of <em>G</em> consisting of all characters of <em>G</em> and <span><math><mover><mrow><mi>f</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>(</mo><mi>χ</mi><mo>)</mo></math></span> is the <em>Fourier coefficient</em> of <em>f</em> at <span><math><mi>χ</mi><mo>∈</mo><mover><mrow><mi>G</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>. In this paper, we show that by performing the fast (inverse) Fourier transform, we are able to compute a sparse Fourier sum of squares (FSOS) certificate of <em>f</em> on a finite abelian group <em>G</em> with complexity that is quasi-linear in the order of <em>G</em> and polynomial in the FSOS sparsity of <em>f</em>. Moreover, for a nonnegative function <em>f</em> on a finite abelian group <em>G</em> and a subset <span><math><mi>S</mi><mo>⊆</mo><mover><mrow><mi>G</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>, we give a lower bound of the constant <em>M</em> such that <span><math><mi>f</mi><mo>+</mo><mi>M</mi></math></span> admits an FSOS supported on <em>S</em>. We demonstrate the efficiency of the proposed algorithm by numerical experiments on various abelian groups of orders up to 10⁷. As applications, we also solve some combinatorial optimization problems and the sum of Hermitian squares (SOHS) problem by sparse FSOS.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"73 ","pages":"Article 101686"},"PeriodicalIF":2.6,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141623688","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}

{"title":"On the accuracy of Prony's method for recovery of exponential sums with closely spaced exponents","authors":"Rami Katz ,&nbsp;Nuha Diab ,&nbsp;Dmitry Batenkov","doi":"10.1016/j.acha.2024.101687","DOIUrl":"10.1016/j.acha.2024.101687","url":null,"abstract":"<div><p>In this paper we establish accuracy bounds of Prony's method (PM) for recovery of sparse measures from incomplete and noisy frequency measurements, or the so-called problem of super-resolution, when the minimal separation between the points in the support of the measure may be much smaller than the Rayleigh limit. In particular, we show that PM is optimal with respect to the previously established min-max bound for the problem, in the setting when the measurement bandwidth is constant, with the minimal separation going to zero. Our main technical contribution is an accurate analysis of the inter-relations between the different errors in each step of PM, resulting in previously unnoticed cancellations. We also prove that PM is numerically stable in finite-precision arithmetic. We believe our analysis will pave the way to providing accurate analysis of known algorithms for the super-resolution problem in full generality.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"73 ","pages":"Article 101687"},"PeriodicalIF":2.6,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141630145","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}

{"title":"Non-separable multidimensional multiresolution wavelets: A Douglas-Rachford approach","authors":"David Franklin ,&nbsp;Jeffrey A. Hogan ,&nbsp;Matthew K. Tam","doi":"10.1016/j.acha.2024.101684","DOIUrl":"10.1016/j.acha.2024.101684","url":null,"abstract":"<div><p>After re-casting the wavelet construction problem as a feasibility problem with constraints arising from the requirements of compact support, smoothness and orthogonality, the Douglas–Rachford algorithm is employed in the search for multi-dimensional, non-separable, compactly supported, smooth, orthogonal, multiresolution wavelets in the case of translations along the integer lattice and isotropic dyadic dilations. An algorithm for the numerical construction of such wavelets is described. By applying the algorithm, new one-dimensional wavelets are produced as well as genuinely non-separable two-dimensional wavelets.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"73 ","pages":"Article 101684"},"PeriodicalIF":2.6,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1063520324000617/pdfft?md5=5f9312281e421d0c1eb84ee1017dcee5&pid=1-s2.0-S1063520324000617-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141630144","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}

{"title":"An unbounded operator theory approach to lower frame and Riesz-Fischer sequences","authors":"Peter Balazs,&nbsp;Mitra Shamsabadi","doi":"10.1016/j.acha.2024.101685","DOIUrl":"10.1016/j.acha.2024.101685","url":null,"abstract":"<div><p>Frames and orthonormal bases are important concepts in functional analysis and linear algebra. They are naturally linked to bounded operators. To describe unbounded operators those sequences might not be well suited. This has already been noted by von Neumann in the 1920ies. But modern frame theory also investigates other sequences, including those that are not naturally linked to bounded operators. The focus of this manuscript will be two such kind of sequences: lower frame and Riesz-Fischer sequences. We will discuss the inter-relation of those sequences. We will fill a hole existing in the literature regarding the classification of these sequences by their synthesis operator. We will use the idea of generalized frame operator and Gram matrix and extend it. We will use that to show properties for canonical duals for lower frame sequences, such as a minimality condition regarding its coefficients. We will also show that other results that are known for frames can be generalized to lower frame sequences. Finally, we show that the converse of a well-known result is true, i.e. that minimal lower frame sequences are equivalent to complete Riesz-Fischer sequences, without any further assumptions.</p><p>To be able to tackle these tasks, we had to revisit the concept of invertibility (in particular for non-closed operators). In addition, we are able to define a particular adjoint, which is uniquely defined for any operator.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"73 ","pages":"Article 101685"},"PeriodicalIF":2.6,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141728875","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}

{"title":"Beurling dimension of spectra for a class of random convolutions on R2","authors":"Jinjun Li,&nbsp;Zhiyi Wu","doi":"10.1016/j.acha.2024.101683","DOIUrl":"https://doi.org/10.1016/j.acha.2024.101683","url":null,"abstract":"<div><p>It is usually difficult to study the structure of the spectra for the measures in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> and higher dimensions. In this paper, by employing the projective techniques and our previous results on the line we prove that the Beurling dimension of spectra for a class of random convolutions in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> satisfies an intermediate value property.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"73 ","pages":"Article 101683"},"PeriodicalIF":2.6,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141594291","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}