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Axioms
Special Issues
Wavelet and Frame Constructions, with Applications
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Wavelet and Frame Constructions, with Applications

A special issue of Axioms (ISSN 2075-1680).

Deadline for manuscript submissions: closed (30 June 2017) | Viewed by 40309

Special Issue Editors

Prof. Dr. Palle E.T. Jorgensen

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Guest Editor
Department of Mathematics, 14 MLH, The University of Iowa, Iowa City, IA 52242-1419, USA
Interests: mathematical physics; Euclidean field theory; reflection positivity; representation theory; operators in Hilbert space; harmonic analysis; fractals; wavelets; stochastic processes; financial mathematics
Special Issues, Collections and Topics in MDPI journals
Dr. Azita Mayeli

E-Mail Website
Guest Editor
Department of Mathematics, The Graduate Center and Queensborough of The City, University of New York (CUNY) 222-05 56th Ave, Bayside, NY 11364, USA
Interests: frame and wavelet theory; spectral theory; representation theory; harmonic analysis; Heisenberg group; exponential bases

Special Issue Information

Dear Colleagues,

We envision a collection of papers in applied and computational harmonic analysis with an emphasis on frame analysis. In technical terms, Hilbert space-frames generalize more standard notions of bases. They are configurations of vectors in Hilbert space, which form overcomplete spanning families; and, hence, they generalize more standard notions of bases. In frame analysis, we give up the orthogonality requirements in ONBs. The introduction and use of frames are dictated by such applications as decomposition of signals, such as images, sound, e.g., music, speech, big data in finance, astrophysics, etc. The advantage of frames over the orthonormal bases is due their flexibility, and their use in reconstruction, multi-band analysis, in sampling theory, and to their ability to account for redundancy and aliasing. In detail, referring to the analysis/synthesis problem: with the use of suitable frames, each Hilbert space vector (signal) can be represented in many different ways in terms of frame elements. This makes the frames very useful in algorithms for recovering of data, for example data lost in Internet coding (so called erasure), in error correction in coding and decoding theory, in minimizing of noise in signal and image processing, and in speech recognition, signal processing, quantum computing, coding theory, and sparse representations, and in electrical engineering. Riesz bases and orthonormal bases are examples of frames.

Frames are constructed on various settings such as Euclidean spaces, manifolds, Lie groups, finite groups, finite fields, graphs and fractals. The structure and role of a frame varies by its application to a given explicit problem, and its adaptation adapts well to the desired setting. Most well known frames are Gabor frames (in connection to signal processing), Grassmannian frames (in connection to the area of spherical code, with application to wireless communication and to multiple description coding), and probabilistic frames as probabilistic measure in relation to statistics, geometry of convex bodies, quantum computing, and compress sensing.

A special kind of Hilbert space frames is wavelets. The construction of wavelet frames takes advantage of similarity up to scale. The functions making up a wavelet frame are obtained by action of a pair of operator families, dilations and translations, and acting on a single function a so-called ``mother function". Wavelets are usually obtained with the use of suitable multiscale analyses; and are popular in applications and decomposition problems due to their efficiency, especially in time-frequency localization problems. The wavelets, as analytic tools, have found a great number of applications in astrophysics, and harmonic analysis, such as in geometric discrepancy theory and in the study of irregularities of distributions.

Prof. Palle E.T. Jorgensen
Dr. Azita Mayeli
Guest Editors

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Axioms is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • frame and wavelet theory
  • Riesz bases
  • exponential bases
  • non-harmonic Fourier series
  • Fuglede conjecture
  • spectral theory
  • representation theory
  • harmonic analysis
  • spherical harmonic analysis
  • operators in Hilbert spaces
  • fractals, finite fields
  • finite groups
  • Lie groups
  • manifolds
  • stochastic processes
  • engineering
  • mathematical physics
  • signal and image processing
  • sampling and interpolation
  • approximation theory
  • compress sensing

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Published Papers (7 papers)

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Research

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19 pages, 805 KiB  
Article
Quincunx Fundamental Refinable Functions in Arbitrary Dimensions
by Xiaosheng Zhuang
Axioms 2017, 6(3), 20; https://doi.org/10.3390/axioms6030020 - 6 Jul 2017
Cited by 2 | Viewed by 3532
Abstract
In this paper, we generalize the family of Deslauriers–Dubuc’s interpolatory masks from dimension one to arbitrary dimensions with respect to the quincunx dilation matrices, thereby providing a family of quincunx fundamental refinable functions in arbitrary dimensions. We show that a family of unique [...] Read more.
In this paper, we generalize the family of Deslauriers–Dubuc’s interpolatory masks from dimension one to arbitrary dimensions with respect to the quincunx dilation matrices, thereby providing a family of quincunx fundamental refinable functions in arbitrary dimensions. We show that a family of unique quincunx interpolatory masks exists and such a family of masks is of real value and has the full-axis symmetry property. In dimension d = 2 , we give the explicit form of such unique quincunx interpolatory masks, which implies the nonnegativity property of such a family of masks. Full article
(This article belongs to the Special Issue Wavelet and Frame Constructions, with Applications)
29 pages, 362 KiB  
Article
Euclidean Algorithm for Extension of Symmetric Laurent Polynomial Matrix and Its Application in Construction of Multiband Symmetric Perfect Reconstruction Filter Bank
by Jianzhong Wang
Axioms 2017, 6(2), 9; https://doi.org/10.3390/axioms6020009 - 20 Apr 2017
Cited by 1 | Viewed by 3898
Abstract
For a given pair of s-dimensional real Laurent polynomials ( a → ( z ) , b → ( z ) ) , which has a certain type of symmetry and satisfies the dual condition [...] Read more.
For a given pair of s-dimensional real Laurent polynomials ( a → ( z ) , b → ( z ) ) , which has a certain type of symmetry and satisfies the dual condition b → ( z ) T a → ( z ) = 1 , an s × s Laurent polynomial matrix A ( z ) (together with its inverse A - 1 ( z ) ) is called a symmetric Laurent polynomial matrix extension of the dual pair ( a → ( z ) , b → ( z ) ) if A ( z ) has similar symmetry, the inverse A - 1 ( Z ) also is a Laurent polynomial matrix, the first column of A ( z ) is a → ( z ) and the first row of A - 1 ( z ) is ( b → ( z ) ) T . In this paper, we introduce the Euclidean symmetric division and the symmetric elementary matrices in the Laurent polynomial ring and reveal their relation. Based on the Euclidean symmetric division algorithm in the Laurent polynomial ring, we develop a novel and effective algorithm for symmetric Laurent polynomial matrix extension. We also apply the algorithm in the construction of multi-band symmetric perfect reconstruction filter banks. Full article
(This article belongs to the Special Issue Wavelet and Frame Constructions, with Applications)
13 pages, 276 KiB  
Article
Fourier Series for Singular Measures
by John E. Herr and Eric S. Weber
Axioms 2017, 6(2), 7; https://doi.org/10.3390/axioms6020007 - 28 Mar 2017
Cited by 13 | Viewed by 5056
Abstract
Using the Kaczmarz algorithm, we prove that for any singular Borel probability measure μ on [ 0 , 1 ) , every f ∈ L 2 ( μ ) possesses a Fourier series of the form [...] Read more.
Using the Kaczmarz algorithm, we prove that for any singular Borel probability measure μ on [ 0 , 1 ) , every f ∈ L 2 ( μ ) possesses a Fourier series of the form f ( x ) = ∑ n = 0 ∞ c n e 2 π i n x . We show that the coefficients c n can be computed in terms of the quantities f ^ ( n ) = ∫ 0 1 f ( x ) e − 2 π i n x d μ ( x ) . We also demonstrate a Shannon-type sampling theorem for functions that are in a sense μ -bandlimited. Full article
(This article belongs to the Special Issue Wavelet and Frame Constructions, with Applications)
15 pages, 400 KiB  
Article
Norm Retrieval and Phase Retrieval by Projections
by Peter G. Casazza, Dorsa Ghoreishi, Shani Jose and Janet C. Tremain
Axioms 2017, 6(1), 6; https://doi.org/10.3390/axioms6010006 - 4 Mar 2017
Cited by 15 | Viewed by 5475
Abstract
We make a detailed study of norm retrieval. We give several classification theorems for norm retrieval and give a large number of examples to go with the theory. One consequence is a new result about Parseval frames: If a Parseval frame is divided [...] Read more.
We make a detailed study of norm retrieval. We give several classification theorems for norm retrieval and give a large number of examples to go with the theory. One consequence is a new result about Parseval frames: If a Parseval frame is divided into two subsets with spans W 1 , W 2 and W 1 ∩ W 2 = { 0 } , then W 1 ⊥ W 2 . Full article
(This article belongs to the Special Issue Wavelet and Frame Constructions, with Applications)
21 pages, 5393 KiB  
Article
Sparse Wavelet Representation of Differential Operators with Piecewise Polynomial Coefficients
by Dana Černá and Václav Finĕk
Axioms 2017, 6(1), 4; https://doi.org/10.3390/axioms6010004 - 22 Feb 2017
Cited by 15 | Viewed by 5130
Abstract
We propose a construction of a Hermite cubic spline-wavelet basis on the interval and hypercube. The basis is adapted to homogeneous Dirichlet boundary conditions. The wavelets are orthogonal to piecewise polynomials of degree at most seven on a uniform grid. Therefore, the wavelets [...] Read more.
We propose a construction of a Hermite cubic spline-wavelet basis on the interval and hypercube. The basis is adapted to homogeneous Dirichlet boundary conditions. The wavelets are orthogonal to piecewise polynomials of degree at most seven on a uniform grid. Therefore, the wavelets have eight vanishing moments, and the matrices arising from discretization of differential equations with coefficients that are piecewise polynomials of degree at most four on uniform grids are sparse. Numerical examples demonstrate the efficiency of an adaptive wavelet method with the constructed wavelet basis for solving the one-dimensional elliptic equation and the two-dimensional Black–Scholes equation with a quadratic volatility. Full article
(This article belongs to the Special Issue Wavelet and Frame Constructions, with Applications)
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15 pages, 250 KiB  
Article
Discrete Frames on Finite Dimensional Left Quaternion Hilbert Spaces
by M. Khokulan, K. Thirulogasanthar and S. Srisatkunarajah
Axioms 2017, 6(1), 3; https://doi.org/10.3390/axioms6010003 - 21 Feb 2017
Cited by 12 | Viewed by 4359
Abstract
An introductory theory of frames on finite dimensional left quaternion Hilbert spaces is demonstrated along the lines of their complex counterpart. Full article
(This article belongs to the Special Issue Wavelet and Frame Constructions, with Applications)

Review

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20 pages, 344 KiB  
Review
The SIC Question: History and State of Play
by Christopher A. Fuchs, Michael C. Hoang and Blake C. Stacey
Axioms 2017, 6(3), 21; https://doi.org/10.3390/axioms6030021 - 18 Jul 2017
Cited by 157 | Viewed by 11022
Abstract
Recent years have seen significant advances in the study of symmetric informationally complete (SIC) quantum measurements, also known as maximal sets of complex equiangular lines. Previously, the published record contained solutions up to dimension 67, and was with high confidence complete up through [...] Read more.
Recent years have seen significant advances in the study of symmetric informationally complete (SIC) quantum measurements, also known as maximal sets of complex equiangular lines. Previously, the published record contained solutions up to dimension 67, and was with high confidence complete up through dimension 50. Computer calculations have now furnished solutions in all dimensions up to 151, and in several cases beyond that, as large as dimension 844. These new solutions exhibit an additional type of symmetry beyond the basic definition of a SIC, and so verify a conjecture of Zauner in many new cases. The solutions in dimensions 68 through 121 were obtained by Andrew Scott, and his catalogue of distinct solutions is, with high confidence, complete up to dimension 90. Additional results in dimensions 122 through 151 were calculated by the authors using Scott’s code. We recap the history of the problem, outline how the numerical searches were done, and pose some conjectures on how the search technique could be improved. In order to facilitate communication across disciplinary boundaries, we also present a comprehensive bibliography of SIC research. Full article
(This article belongs to the Special Issue Wavelet and Frame Constructions, with Applications)
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