735 results on '"Fourier inversion theorem"'
Search Results
2. Explicit inversion of Band Toeplitz matrices by discrete Fourier transform
- Author
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Mohamed Elouafi
- Subjects
Algebra and Number Theory ,Discrete-time Fourier transform ,010102 general mathematics ,Mathematical analysis ,Fourier inversion theorem ,010103 numerical & computational mathematics ,01 natural sciences ,Fractional Fourier transform ,Toeplitz matrix ,symbols.namesake ,Fourier transform ,Discrete sine transform ,Discrete Fourier series ,symbols ,0101 mathematics ,Fourier transform on finite groups ,Mathematics - Abstract
In this paper, we give an explicit formula for the element of the inverse where is a band Toeplitz matrix with left bandwidth s and right bandwidth r. The formula involves determinants, , whose elements are the discrete Fourier transform of where f is the symbol of .
- Published
- 2017
3. On the matrix Fourier filtering problem for a class of models of nonlinear optical systems with a feedback
- Author
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A. V. Razgulin and S. V. Sazonova
- Subjects
DFT matrix ,Discrete-time Fourier transform ,Fourier inversion theorem ,Mathematical analysis ,01 natural sciences ,010101 applied mathematics ,010309 optics ,Computational Mathematics ,symbols.namesake ,Fourier analysis ,Discrete Fourier series ,0103 physical sciences ,symbols ,0101 mathematics ,Fourier series ,Mathematics ,Fourier transform on finite groups ,Sine and cosine transforms - Abstract
A novel statement of the Fourier filtering problem based on the use of matrix Fourier filters instead of conventional multiplier filters is considered. The basic properties of the matrix Fourier filtering for the filters in the Hilbert–Schmidt class are established. It is proved that the solutions with a finite energy to the periodic initial boundary value problem for the quasi-linear functional differential diffusion equation with the matrix Fourier filtering Lipschitz continuously depend on the filter. The problem of optimal matrix Fourier filtering is formulated, and its solvability for various classes of matrix Fourier filters is proved. It is proved that the objective functional is differentiable with respect to the matrix Fourier filter, and the convergence of a version of the gradient projection method is also proved.
- Published
- 2017
4. Compact Fractional Fourier Domains
- Author
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Ahmet Serbes
- Subjects
Discrete-time Fourier transform ,Non-uniform discrete Fourier transform ,Applied Mathematics ,Fourier inversion theorem ,Mathematical analysis ,020206 networking & telecommunications ,02 engineering and technology ,01 natural sciences ,Fractional Fourier transform ,010309 optics ,symbols.namesake ,Fourier transform ,Fourier analysis ,Discrete Fourier series ,0103 physical sciences ,Signal Processing ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,Electrical and Electronic Engineering ,Fourier series ,Mathematics - Abstract
In this letter, a method for finding the compact fractional Fourier domains is presented. As a measure of compactness, we take the $\ell _1$ -norm. It is proposed that there exists at least one fractional Fourier domain in which $\ell _1$ -norm of the transformed signal is minimum. A coarse-to-fine grid search strategy is adopted to find the optimum fractional Fourier transform angle that makes $\ell _1$ -norm minimum with low computational cost. Extensive simulation results validate the proposed method.
- Published
- 2017
5. Estimate of the Fourier multipliers in the spherical mean setting
- Author
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S. Omri and B. Majjaouli
- Subjects
Discrete-time Fourier transform ,Non-uniform discrete Fourier transform ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Fourier inversion theorem ,01 natural sciences ,Spherical mean ,010101 applied mathematics ,symbols.namesake ,Fourier analysis ,Discrete Fourier series ,symbols ,0101 mathematics ,Fourier series ,Analysis ,Sine and cosine transforms ,Mathematics - Published
- 2016
6. Convergence of Fourier series with respect to general orthonormal systems
- Author
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V. Sh. Tsagareishvili and Larry Gogoladze
- Subjects
0209 industrial biotechnology ,General Mathematics ,010102 general mathematics ,Function series ,Fourier inversion theorem ,Wavelet transform ,02 engineering and technology ,01 natural sciences ,020901 industrial engineering & automation ,Generalized Fourier series ,Control theory ,Discrete Fourier series ,Conjugate Fourier series ,Applied mathematics ,Orthonormal basis ,0101 mathematics ,Fourier series ,Mathematics - Published
- 2016
7. Alternative Fourier Series Expansions with Accelerated Convergence
- Author
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Wenlong Li
- Subjects
Mathematical analysis ,Fourier inversion theorem ,Function series ,010103 numerical & computational mathematics ,General Medicine ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Generalized Fourier series ,Fourier analysis ,Discrete Fourier series ,Conjugate Fourier series ,symbols ,0101 mathematics ,Fourier series ,Sine and cosine transforms ,Mathematics - Abstract
The key objective of this paper is to improve the approximation of a sufficiently smooth nonperiodic function defined on a compact interval by proposing alternative forms of Fourier series expansions. Unlike in classical Fourier series, the expansion coefficients herein are explicitly dependent not only on the function itself, but also on its derivatives at the ends of the interval. Each of these series expansions can be made to converge faster at a desired polynomial rate. These results have useful implications to Fourier or harmonic analysis, solutions to differential equations and boundary value problems, data compression, and so on.
- Published
- 2016
8. Fourier Series in Weighted Lorentz Spaces
- Author
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Gord Sinnamon and Javad Rastegari
- Subjects
Bispinor ,Physics::General Physics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Fourier inversion theorem ,010103 numerical & computational mathematics ,Lorentz covariance ,Physics::Classical Physics ,01 natural sciences ,Velocity-addition formula ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,symbols.namesake ,Lorentz space ,Fourier analysis ,Discrete Fourier series ,FOS: Mathematics ,symbols ,Primary 42B35, Secondary 46E30, 42B05 ,0101 mathematics ,Fourier series ,Analysis ,Mathematics - Abstract
The Fourier coefficient map is considered as an operator from a weighted Lorentz space on the circle to a weighted Lorentz sequence space. For a large range of Lorentz indices, necessary and sufficient conditions on the weights are given for the map to be bounded. In addition, new direct analogues are given for known weighted Lorentz space inequalities for the Fourier transform. Applications are given that involve Fourier coefficients of functions in LlogL and more general Lorentz-Zygmund spaces., Comment: The final publication is available at Springer via http://dx.doi.org/10.1007/s00041-015-9455-5
- Published
- 2015
9. Application of the least-squares inversion method: Fourier series versus waveform inversion
- Author
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Jungkyun Shin, Dong-Joo Min, and Changsoo Shin
- Subjects
Hessian matrix ,symbols.namesake ,Geophysics ,Fourier analysis ,Discrete Fourier series ,Diagonal matrix ,Fourier inversion theorem ,Mathematical analysis ,symbols ,Waveform ,Fourier series ,Sine and cosine transforms ,Mathematics - Abstract
We describe an implicit link between waveform inversion and Fourier series based on inversion methods such as gradient, Gauss–Newton, and full Newton methods. Fourier series have been widely used as a basic concept in studies on seismic data interpretation, and their coefficients are obtained in the classical Fourier analysis. We show that Fourier coefficients can also be obtained by inversion algorithms, and compare the method to seismic waveform inversion algorithms. In that case, Fourier coefficients correspond to model parameters (velocities, density or elastic constants), whereas cosine and sine functions correspond to components of the Jacobian matrix, that is, partial derivative wavefields in seismic inversion. In the classical Fourier analysis, optimal coefficients are determined by the sensitivity of a given function to sine and cosine functions. In the inversion method for Fourier series, Fourier coefficients are obtained by measuring the sensitivity of residuals between given functions and test functions (defined as the sum of weighted cosine and sine functions) to cosine and sine functions. The orthogonal property of cosine and sine functions makes the full or approximate Hessian matrix become a diagonal matrix in the inversion for Fourier series. In seismic waveform inversion, the Hessian matrix may or may not be a diagonal matrix, because partial derivative wavefields correlate with each other to some extent, making them semi-orthogonal. At the high-frequency limits, however, the Hessian matrix can be approximated by either a diagonal matrix or a diagonally-dominant matrix. Since we usually deal with relatively low frequencies in seismic waveform inversion, it is not diagonally dominant and thus it is prohibitively expensive to compute the full or approximate Hessian matrix. By interpreting Fourier series with the inversion algorithms, we note that the Fourier series can be computed at an iteration step using any inversion algorithms such as the gradient, full-Newton, and Gauss–Newton methods similar to waveform inversion.
- Published
- 2015
10. The Lerch transcendent from the point of view of Fourier analysis
- Author
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Francisco J. Ruiz, Luis M. Navas, and Juan L. Varona
- Subjects
Polylogarithm ,Bernoulli polynomials ,Mathematics::Number Theory ,Applied Mathematics ,Mathematical analysis ,Fourier inversion theorem ,Hurwitz zeta functions ,Lerch transcendent function ,Fourier series ,Hurwitz zeta function ,symbols.namesake ,Fourier analysis ,Discrete Fourier series ,Conjugate Fourier series ,symbols ,Analysis ,Mathematics - Abstract
We obtain some well-known expansions for the Lerch transcendent and the Hurwitz zeta function using elementary Fourier analytic methods. These Fourier series can be used to analytically continue the functions and prove the classical functional equations, which arise from the relations satisfied by the Fourier conjugate and flat Fourier series. In particular, the functional equation for the Riemann zeta function can be obtained in this way without contour integrals. The conjugate series for special values of the parameters yields analogous results for the Bernoulli and Apostol-Bernoulli polynomials. Finally, we give some consequences derived from the Fourier series. © 2015 Elsevier Inc.
- Published
- 2015
11. Almost Periodic Measures and their Fourier Transforms
- Author
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Nicolae Strungaru and Robert V. Moody
- Subjects
Physics ,symbols.namesake ,Fourier transform ,Fourier analysis ,Phase correlation ,Discrete Fourier series ,Mathematical analysis ,Fourier inversion theorem ,symbols ,Fourier series ,Sine and cosine transforms - Published
- 2017
12. Fourier Pairs of Discrete Support with Little Structure
- Author
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Mihail N. Kolountzakis
- Subjects
Discrete mathematics ,Discrete-time Fourier transform ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Poisson summation formula ,Fourier inversion theorem ,Discrete measure ,01 natural sciences ,symbols.namesake ,Discrete Fourier transform (general) ,Discrete sine transform ,Mathematics - Classical Analysis and ODEs ,Discrete Fourier series ,0103 physical sciences ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,symbols ,010307 mathematical physics ,0101 mathematics ,Analysis ,Fourier transform on finite groups ,Mathematics - Abstract
We give a simple proof of the fact that there exist measures on the real line of discrete support, whose Fourier Transform is also a measure of discrete support, yet this Fourier pair cannot be constructed by repeatedly applying the Poisson Summation Formula finitely many times. More specifically the support of both the measure and its Fourier Tranform are not contained in a finite union of arithmetic progressions., The bibliography was missing from the previous version
- Published
- 2015
13. On multiple Fourier coefficients of a function of the generalized Wiener class
- Author
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R. G. Vyas
- Subjects
symbols.namesake ,Fourier transform ,Generalized Fourier series ,Fourier analysis ,General Mathematics ,Discrete Fourier series ,Mathematical analysis ,Fourier inversion theorem ,symbols ,Fourier series ,Fractional Fourier transform ,Sine and cosine transforms ,Mathematics - Abstract
In this paper, we estimate the order of magnitude of the double Fourier coefficients of a function of the class ( Λ 1 , Λ 2 ) BV ( p ( n ) ↑ ∞ , ϕ ) ${(\Lambda ^1,\Lambda ^2)\operatorname{BV}(p(n)\uparrow \infty ,\varphi )}$ over [ 0 , 2 π ] 2 ${[0,2\pi ]^2}$ .
- Published
- 2015
14. On the Equiconvergence of the Fourier Series and Integral of Distributions
- Author
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Abdumalik Rakhimov
- Subjects
Mathematics::Functional Analysis ,Discrete Fourier series ,Fourier sine and cosine series ,Mathematical analysis ,Fourier inversion theorem ,Mathematics::Classical Analysis and ODEs ,Mathematics::Spectral Theory ,Fourier series ,Mathematics - Abstract
We prove equiconvergence of the Bochner-Riesz means of the Fourier series and integral of distributions with compact support from the Liouville spaces.
- Published
- 2015
15. Further Discussion on the Calculation of Fourier Series
- Author
-
Caixia Zhang
- Subjects
Generalized Fourier series ,Discrete Fourier series ,Function series ,Mathematical analysis ,Conjugate Fourier series ,Fourier sine and cosine series ,Fourier inversion theorem ,General Medicine ,Fourier series ,Sine and cosine transforms ,Mathematics - Abstract
Fourier series is an important mathematical concept. It is well known that we need too much computation to expand the function into Fourier series. The existing literature only pointed that its Fourier series is sine series when the function is an odd function and its Fourier series is cosine series when the function is an even function. And on this basis, in this paper, according to the function which satisfies different conditions, we give the different forms of Fourier series and the specific calculation formula of Fourier coefficients, so as to avoid unnecessary calculation. In addition, if a function is defined on [0,a], we can make it have some kind of nature by using the extension method as needed. So we can get the corresponding form of Fourier series.
- Published
- 2015
16. Fourier transformation of quasiconvex functions and functions of the class V *
- Author
-
Roald Mikhailovich Trigub
- Subjects
Statistics and Probability ,Uses of trigonometry ,Applied Mathematics ,General Mathematics ,Fourier inversion theorem ,Mathematical analysis ,Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) ,Parseval's theorem ,symbols.namesake ,Fourier transform ,Computer Science::Sound ,Fourier analysis ,Discrete Fourier series ,Conjugate Fourier series ,symbols ,Fourier series ,Mathematics - Abstract
One Szokefalvi-Nagy theorem is utterly strengthened, and a new relationship between Fourier series and integrals is established.
- Published
- 2014
17. Algorithmic Approach for Formal Fourier Series
- Author
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Etienne Nana Chiadjeu and Wolfram Koepf
- Subjects
Uses of trigonometry ,Applied Mathematics ,Fourier inversion theorem ,Fourier sine and cosine series ,Trigonometric polynomial ,Algebra ,Computational Mathematics ,symbols.namesake ,Computational Theory and Mathematics ,Fourier analysis ,Discrete Fourier series ,symbols ,Fourier series ,Sine and cosine transforms ,Mathematics - Abstract
The study of trigonometric series has started at the beginning of the nineteenth century. Joseph Fourier made the important observation that almost every function of a closed interval can be decomposed into the sum of sine and cosine functions. This technique to develop a function into a trigonometric series was published for the first time in 1822 by Joseph Fourier. The resulting series is nowadays called Fourier series. Since Fourier’s time, many different approaches to understand the concept of Fourier series have been discovered, each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on mathematical ideas and tools that were not available at the time Fourier completed his original work. Although the original motivation was to solve the heat equation for a metal plate, it later became obvious that the same technique could be applied to a wide variety of mathematical and physical problems and has many applications in electrical engineering, vibration analysis, acoustics, optics, signal treatment, image processing, etc. Despite the importance of Fourier series, the method used until now to compute them via computer algebra systems (CAS) is essentially based on the same principle as in Fourier’s time, i.e. by the evaluation of certain integrals. Unfortunately this technique is not completely successful for many functions. Although numeric values of the Fourier coefficients might be available, symbolic values are often not accessible. Modern CAS like Maple or Mathematica can compute such integrals in many cases for a given \({n \in \mathbb{Z}}\). However if one is interested in the Fourier coefficients for all \({n \in \mathbb{Z}}\), then n is considered as a given symbolic variable and such integrals can be computed only in few cases. In this paper we introduce an algorithmic approach to compute those Fourier coefficients, involving differential equations of a particular form, and recurrence equations. This approach extrapolates the computation of the Fourier series for functions for which the computation of Fourier coefficients via the definition is out of reach for current CAS.
- Published
- 2014
18. On the Reduced Noise Sensitivity of a New Fourier Transformation Algorithm
- Author
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H. Szegedi, Judit Molnár, Péter Szűcs, and Mihály Dobróka
- Subjects
Discrete-time Fourier transform ,Non-uniform discrete Fourier transform ,Fourier inversion theorem ,Short-time Fourier transform ,Fractional Fourier transform ,symbols.namesake ,Mathematics (miscellaneous) ,Fourier transform ,Fourier analysis ,Discrete Fourier series ,symbols ,General Earth and Planetary Sciences ,Algorithm ,Mathematics - Abstract
In this study, a new inversion method is presented for performing one-dimensional Fourier transform, which shows highly robust behavior against noises. As the Fourier transformation is linear, the data noise is also transformed to the frequency domain making the operation noise sensitive especially in case of non-Gaussian noise distribution. In the field of inverse problem theory it is well known that there are numerous procedures for noise rejection, so if the Fourier transformation is formulated as an inverse problem these tools can be used to reduce the noise sensitivity. It was demonstrated in many case studies that the method of most frequent value provides useful weights to increase the noise rejection capability of geophysical inversion methods. Following the basis of the latter method the Fourier transform is formulated as an iteratively reweighted least squares problem using Steiner’s weights. Series expansion was applied to the discretization of the continuous functions of the complex spectrum. It is shown that the Jacobian matrix of the inverse problem can be calculated as the inverse Fourier transform of the basis functions used in the series expansion. To avoid the calculation of the complex integral a set of basis functions being eigenfunctions of the inverse Fourier transform is produced. This procedure leads to the modified Hermite functions and results in quick and robust inversion-based Fourier transformation method. The numerical tests of the procedure show that the noise sensitivity can be reduced around an order of magnitude compared to the traditional discrete Fourier transform.
- Published
- 2014
19. Computation of two-dimensional Fourier transforms for noisy band-limited signals
- Author
-
Weidong Chen
- Subjects
Discrete-time Fourier transform ,Non-uniform discrete Fourier transform ,Applied Mathematics ,Mathematical analysis ,Fourier inversion theorem ,Fractional Fourier transform ,Discrete Fourier transform ,Computational Mathematics ,symbols.namesake ,Fourier analysis ,Discrete Fourier series ,symbols ,Fourier series ,Mathematics - Abstract
In this paper, the ill-posedness of computing the two dimensional Fourier transform is discussed. A regularized algorithm for computing the two dimensional Fourier transform of band-limited signals is presented. The convergence of the regularized Fourier series is studied and compared with the Fourier series by some examples.
- Published
- 2014
20. Appraisal problem in the 3D least squares Fourier seismic data reconstruction
- Author
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Nicola Bienati, Alfredo Mazzotti, Fabio Ciabarri, and Eusebio Stucchi
- Subjects
Least-squares spectral analysis ,Discrete-time Fourier transform ,Non-uniform discrete Fourier transform ,Fourier inversion theorem ,Geophysics ,Discrete Fourier transform ,Fractional Fourier transform ,symbols.namesake ,Geochemistry and Petrology ,Fourier analysis ,Discrete Fourier series ,symbols ,Algorithm ,Geology - Abstract
Least squares Fourier reconstruction is basically a solution to a discrete linear inverse problem that attempts to recover the Fourier spectrum of the seismic wavefield from irregularly sampled data along the spatial coordinates. The estimated Fourier coefficients are then used to reconstruct the data in a regular grid via a standard inverse Fourier transform (inverse discrete Fourier transform or inverse fast Fourier transform). Unfortunately, this kind of inverse problem is usually under-determined and illconditioned. For this reason, the least squares Fourier reconstruction with minimum norm adopts a damped least squares inversion to retrieve a unique and stable solution. In this work, we show how the damping can introduce artefacts on the reconstructed 3D data. To quantitatively describe this issue, we introduce the concept of “extended” model resolution matrix, and we formulate the reconstruction problem as an appraisal problem. Through the simultaneous analysis of the extended model resolution matrix and of the noise term, we discuss the limits of the Fourier reconstruction with minimum norm reconstruction and assess the validity of the reconstructed data and the possible bias introduced by the inversion process. Also, we can guide the parameterization of the forward problem to minimize the occurrence of unwanted artefacts. A simple synthetic example and real data from a 3D marine common shot gather are used to discuss our approach and to show the results of Fourier reconstruction with minimum norm reconstruction.
- Published
- 2014
21. Fast sparse nonlinear Fourier expansions of high dimensional functions
- Author
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Rui Wang, Haiye Yu, and Xu Liu
- Subjects
TheoryofComputation_MISCELLANEOUS ,Discrete-time Fourier transform ,Non-uniform discrete Fourier transform ,Applied Mathematics ,Fourier inversion theorem ,Mathematical analysis ,MathematicsofComputing_NUMERICALANALYSIS ,Split-step method ,symbols.namesake ,Fourier analysis ,Discrete Fourier series ,symbols ,Pseudo-spectral method ,Fourier series ,Analysis ,Mathematics - Abstract
The nonlinear Fourier basis has shown its advantages over the classical Fourier basis in the time-frequency analysis. The need of processing large amount of high dimensional data motivates the extension of the methods based upon the nonlinear Fourier basis to high dimensions. We consider the multi-dimensional nonlinear Fourier basis, which is the tensor product of univariate nonlinear Fourier basis. We investigate the convergence order in norm and also the almost everywhere convergence of the nonlinear Fourier expansions. In order to compute fast and efficiently the nonlinear Fourier expansions of d-dimensional functions, we introduce the sparse nonlinear Fourier expansion and develop a fast algorithm for evaluating it. We also prove that the fast sparse nonlinear Fourier expansions enjoy the optimal convergence order and reduce the computational costs to O ( n log 2 d − 1 n ) . Numerical experiments are presented to demonstrate the efficiency and accuracy of the proposed method.
- Published
- 2014
22. Linear Inversion in Fourier Space
- Author
-
Natalia K. Nikolova
- Subjects
symbols.namesake ,Fourier transform ,Computer science ,Discrete-time Fourier transform ,Non-uniform discrete Fourier transform ,Discrete Fourier series ,Phase correlation ,Fourier inversion theorem ,symbols ,Iterative reconstruction ,Algorithm ,Fractional Fourier transform - Abstract
In this chapter, methods for image reconstruction, both qualitative and quantitative, are described that solve a linearized model of scattering in Fourier space. They are referred to as spectral-domain reconstruction methods. In all of these methods, the data sets, which are originally functions of the observation position in real space (e.g., x and y ), are subjected to Fourier transformation. The reconstruction of the object's contrast function is then performed in Fourier space, e.g., with respect to k x and k y (the Fourier variables corresponding to x and y ). This necessitates a final step of an inverse Fourier transform to return the result in real space. This category of reconstruction methods includes MW holography, diffraction tomography, and a great variety of techniques used to produce imagery from synthetic aperture radar (SAR) measurements. The spectral-domain reconstruction methods are well studied. Their greatest advantage is that they are fast, allowing for image generation in real time. Their disadvantage is that, like other linear inversion methods, they are limited to problems where multiplescattering and mutual-coupling effects in the object under test (OUT) can be neglected. They are the workhorse of the real-time MW and millimeter-wave imaging systems with various applications in concealed weapon detection [62, 63, 164, 165, 166], nondestructive testing [167–173], medical-imaging research [174–179], antenna measurements [180, 181], and many other areas. The subject of reconstruction in Fourier space is extensive, and it would be impossible to present it in its entirety here. The goal here is to introduce its basic principles and to give the reader an appreciation for the mathematical beauty of the methodology and its power in real-life applications. MW holography is chosen as the method through which the reader is introduced to the subject. There are several reasons for this choice. First, modern MW holography is akin to SAR reconstruction; in fact, it was developed as an extension of SAR to 3D imaging with data acquired on planes. Thus, it can serve as a sound basis for further studies of the various SAR imaging methods. Second, when applied in a cylindrical coordinate system, MW holography can also be viewed as an extension of diffraction tomography to 3D imaging. Third, MW holography is arguably the most widely used real-time reconstruction method applied to data in the MW and millimeter-wave frequency ranges.
- Published
- 2017
23. Fourier extension and sampling on the sphere
- Author
-
Niel Van Buggenhout and Daniel Potts
- Subjects
Discrete-time Fourier transform ,010102 general mathematics ,Fourier inversion theorem ,Fourier sine and cosine series ,Mathematical analysis ,010103 numerical & computational mathematics ,01 natural sciences ,symbols.namesake ,Fourier transform ,Fourier analysis ,Discrete Fourier series ,symbols ,0101 mathematics ,Fourier series ,Mathematics ,Sine and cosine transforms - Abstract
We present different sampling methods for the approximation of functions on the sphere. In this note we focus on Fourier methods on the sphere based on spherical harmonics and on the double Fourier sphere method. Further longitude-latitude transformation is combined with Fourier extension to allow the use of bi-periodic Fourier series on the sphere. Fourier extension with hermite interpolation is introduced and double Fourier sphere method is discussed shortly.
- Published
- 2017
24. On sampling theorems for fractional Fourier transforms and series
- Author
-
Ahmed I. Zayed
- Subjects
Pure mathematics ,Mathematical analysis ,Fourier inversion theorem ,020206 networking & telecommunications ,02 engineering and technology ,01 natural sciences ,Discrete Fourier transform ,Fractional Fourier transform ,010101 applied mathematics ,symbols.namesake ,Fourier transform ,Fourier analysis ,Discrete Fourier series ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,0101 mathematics ,Fourier series ,Sine and cosine transforms ,Mathematics - Abstract
We introduce two generalizations of the exponential function and, hence, two generalizations of the Fourier transform and series. Each generalization depends on a real, nonnegative parameter less than or equal to one, but reduces to the standard exponential function when the parameter is equal to one. For this reason, the corresponding transforms are called fractional Fourier transforms. In this talk we examine sampling theorems of bandlimited functions in these two fractional Fourier transform domains.
- Published
- 2017
25. On almost everywhere covergence of dyadic Fourier series in L2
- Author
-
Anvarjon A. Ahmedov and F. Deraman
- Subjects
Mathematics::Functional Analysis ,Logarithm ,Function series ,Mathematical analysis ,Fourier inversion theorem ,Mathematics::Classical Analysis and ODEs ,Quantitative Biology::Other ,Dirichlet kernel ,symbols.namesake ,Discrete Fourier series ,symbols ,Almost everywhere ,Wu's method of characteristic set ,Fourier series ,Mathematics - Abstract
The almost everywhere convergence of the dyadic Fourier series in L2 is studied. The logarithmic behaviour of the partial sums of Dyadic Fourier series in L2 is established. In order to obtain the estimation for the maximal operator corresponding to the dyadic Fourier series, the properties and asymptotical behaviour of the Dirichlet kernel are investigated. The general representation in the dyadic group and the properties of the characteristic set are used.
- Published
- 2017
26. Approximation of Functions on a Square by Interpolation Polynomials at Vertices and Few Fourier Coefficients
- Author
-
Zhihua Zhang
- Subjects
Article Subject ,lcsh:Mathematics ,010102 general mathematics ,Fourier inversion theorem ,Mathematical analysis ,010103 numerical & computational mathematics ,Linear interpolation ,Trigonometric polynomial ,lcsh:QA1-939 ,01 natural sciences ,symbols.namesake ,Fourier analysis ,Discrete Fourier series ,symbols ,0101 mathematics ,Fourier series ,Analysis ,Interpolation ,Mathematics ,Trigonometric interpolation - Abstract
For a bivariate function on a square, in general, its Fourier coefficients decay slowly, so one cannot reconstruct it by few Fourier coefficients. In this paper we will develop a new approximation scheme to overcome the weakness of Fourier approximation. In detail, we will use Lagrange interpolation and linear interpolation on the boundary of the square to derive a new approximation scheme such that we can use the values of the target function at vertices of the square and few Fourier coefficients to reconstruct the target function with very small error.
- Published
- 2017
27. Fourier Series and Transform
- Author
-
Xin-She Yang
- Subjects
symbols.namesake ,Fourier analysis ,Discrete-time Fourier transform ,Computer science ,Discrete Fourier series ,Fourier inversion theorem ,Fourier sine and cosine series ,symbols ,Discrete transform ,Fourier series ,Algorithm ,Sine and cosine transforms - Abstract
Fourier series and fast Fourier transforms have important applications in signal and image processing. This chapter first introduces Fourier series and then Fourier transforms.
- Published
- 2017
28. The Billard Theorem for Multiple Random Fourier Series
- Author
-
Lionel Moisan, Samuel Ronsin, Hermine Biermé, Mathématiques Appliquées Paris 5 (MAP5 - UMR 8145), Université Paris Descartes - Paris 5 (UPD5)-Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques et Applications (LMA-Poitiers), Université de Poitiers-Centre National de la Recherche Scientifique (CNRS), Mathématiques Appliquées à Paris 5 ( MAP5 - UMR 8145 ), Université Paris Descartes - Paris 5 ( UPD5 ) -Institut National des Sciences Mathématiques et de leurs Interactions-Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Mathématiques et Applications ( LMA-Poitiers ), Université de Poitiers-Centre National de la Recherche Scientifique ( CNRS ), MAP5 - Mathématiques Appliquées à Paris 5 (MAP5), and Université Paris Descartes - Paris 5 (UPD5) - Institut National des Sciences Mathématiques et de leurs Interactions - Centre National de la Recherche Scientifique (CNRS)
- Subjects
[MATH.MATH-PR] Mathematics [math]/Probability [math.PR] ,Multiple Fourier series ,Discrete-time Fourier transform ,General Mathematics ,02 engineering and technology ,Random phase ,01 natural sciences ,Parseval's theorem ,symbols.namesake ,0202 electrical engineering, electronic engineering, information engineering ,Applied mathematics ,0101 mathematics ,Fourier series ,Mathematics ,Random field ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Fourier inversion theorem ,Billard Theorem ,16. Peace & justice ,MSC 2010 Primary: 42B05, 60G60, 60G17 ,Secondary: 42B08, 60G50 ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Fourier analysis ,Discrete Fourier series ,Conjugate Fourier series ,symbols ,020201 artificial intelligence & image processing ,Random fields ,[ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR] ,Analysis ,Random Fourier series - Abstract
International audience; We propose a generalization of a classical result on random Fourier series, namely the Billard Theorem, for random Fourier series over the d-dimensional torus. We provide an investigation of the independence with respect to a choice of a sequence of partial sums (or method of summation). We also study some probabilistic properties of the resulting sum field such as stationarity and characteristics of the marginal distribution.
- Published
- 2017
29. The Discrete Fourier Transform
- Author
-
Tim Olson
- Subjects
Discrete mathematics ,symbols.namesake ,Discrete Fourier transform (general) ,Fourier transform ,Discrete-time Fourier transform ,Discrete Fourier series ,Fourier inversion theorem ,symbols ,Fourier series ,Mathematics ,Parseval's theorem ,Fourier transform on finite groups - Abstract
Fourier Series is a way to represent a function \(f(t) \in L^2[a, b]\) of a continuous variable t with a countable number of coefficients \(c_k\). Oftentimes, however, we are interested in representing a finite number of data points \(\{f(t_k)\}_{k=0}^{N-1}\) which probably come as samples of a function of a continuous variable t. We may not have enough information to represent the original function, but we would still like to know what its Fourier Transform looks like. We have three primary goals for our discrete Fourier Analysis: (1) having an accurate representation, (2) being able to calculate the representation quickly and easily, and (3) knowing what the coefficients of that representation represent. We will now try to accomplish these goals.
- Published
- 2017
30. Improving convergence for the approximation of non-periodic functions by Fourier series
- Author
-
R. V. Golovanov and N. N. Kalitkin
- Subjects
Approximation theory ,Smoothness ,Mathematical analysis ,Fourier inversion theorem ,Trigonometric polynomial ,Periodic function ,Computational Mathematics ,symbols.namesake ,Fourier analysis ,Modeling and Simulation ,Discrete Fourier series ,symbols ,Fourier series ,Mathematics - Abstract
Approximation of functions by Fourier series plays an important role in many applied problems of digital signal processing. An effective method is presented for the construction of highly accurate mean-square approximations by Fourier series for nonperiodic functions. This technique employs the subtraction of specially selected functions that enhance the smoothness of the periodic extension of the approximated function. The main advantage of the method is that the function-setting interval is taken as a half-period rather than a whole period. This doubles the smoothness of the periodic extension. The efficiency of the method is illustrated by test functions of one and two variables.
- Published
- 2014
31. Stability and spectral convergence of Fourier method for nonlinear problems: on the shortcomings of the $$2/3$$ 2 / 3 de-aliasing method
- Author
-
Eitan Tadmor and Claude Bardos
- Subjects
Uses of trigonometry ,Applied Mathematics ,Mathematical analysis ,Fourier inversion theorem ,Split-step method ,Computational Mathematics ,symbols.namesake ,Fourier analysis ,Discrete Fourier series ,symbols ,Pseudo-spectral method ,Spectral method ,Fourier series ,Mathematics - Abstract
The high-order accuracy of Fourier method makes it the method of choice in many large scale simulations. We discuss here the stability of Fourier method for nonlinear evolution problems, focusing on the two prototypical cases of the inviscid Burgers' equation and the multi-dimensional incompressible Euler equations. The Fourier method for such problems with quadratic nonlinearities comes in two main flavors. One is the spectral Fourier method. The other is the $$2/3$$ 2 / 3 pseudo-spectral Fourier method, where one removes the highest $$1/3$$ 1 / 3 portion of the spectrum; this is often the method of choice to maintain the balance of quadratic energy and avoid aliasing errors. Two main themes are discussed in this paper. First, we prove that as long as the underlying exact solution has a minimal $$C^{1+\alpha }$$ C 1 + ? spatial regularity, then both the spectral and the $$2/3$$ 2 / 3 pseudo-spectral Fourier methods are stable. Consequently, we prove their spectral convergence for smooth solutions of the inviscid Burgers equation and the incompressible Euler equations. On the other hand, we prove that after a critical time at which the underlying solution lacks sufficient smoothness, then both the spectral and the $$2/3$$ 2 / 3 pseudo-spectral Fourier methods exhibit nonlinear instabilities which are realized through spurious oscillations. In particular, after shock formation in inviscid Burgers' equation, the total variation of bounded (pseudo-) spectral Fourier solutions must increase with the number of increasing modes and we stipulate the analogous situation occurs with the 3D incompressible Euler equations: the limiting Fourier solution is shown to enforce $$L^2$$ L 2 -energy conservation, and the contrast with energy dissipating Onsager solutions is reflected through spurious oscillations.
- Published
- 2014
32. Fourier series of functions with infinite discontinuities
- Author
-
Branko Sarić
- Subjects
Harmonic analysis ,symbols.namesake ,Sigma approximation ,Fourier analysis ,Discrete Fourier series ,Fourier inversion theorem ,Conjugate Fourier series ,Mathematical analysis ,symbols ,General Medicine ,Classification of discontinuities ,Fourier series ,Mathematics - Abstract
Using the total H1-integrability concept we shall show that functions, which take on infinite values in the interval (−π, π) at only finitely many places, can be expanded into a Fourier series over this interval.
- Published
- 2014
33. Inversion of the continuous windowed Fourier transform using discrete series
- Author
-
WenChang Sun
- Subjects
symbols.namesake ,Fourier transform ,Fourier analysis ,Non-uniform discrete Fourier transform ,Discrete-time Fourier transform ,General Mathematics ,Discrete Fourier series ,Mathematical analysis ,Fourier inversion theorem ,symbols ,Fourier series ,Fractional Fourier transform ,Mathematics - Abstract
We study the inversion formula for the continuous windowed Fourier transform. Different from the classical ones where a single or double integral is involved, we show that for a large class of window functions, a function can be reconstructed from its continuous windowed Fourier transform with a discrete series. Moreover, we show that the series is convergent almost everywhere on R as well as in L p (R) if the function to be reconstructed is. In particular, for the case of p = 2, we give a necessary and sufficient condition for the series to be convergent to the original function.
- Published
- 2014
34. On the Absolute Summability of Fourier Series of Almost Periodic Functions
- Author
-
Yu. Kh. Khasanov
- Subjects
Almost periodic function ,General Mathematics ,media_common.quotation_subject ,Mathematical analysis ,Fourier inversion theorem ,Spectrum (functional analysis) ,Mathematics::Classical Analysis and ODEs ,Infinity ,symbols.namesake ,Fourier analysis ,Discrete Fourier series ,Limit point ,symbols ,Fourier series ,media_common ,Mathematics - Abstract
We establish new sufficient conditions for the absolute |C, α|-summability of the Fourier series of functions almost periodic in a sense of Besicovitch whose spectrum has limit points at infinity and at the origin for $$ \alpha \ge \frac{1}{2} $$ .
- Published
- 2014
35. On the Numerical Stability of Fourier Extensions
- Author
-
Jesús Martín-Vaquero, Ben Adcock, and Daan Huybrechs
- Subjects
Uses of trigonometry ,Discrete-time Fourier transform ,Applied Mathematics ,Mathematical analysis ,Fourier inversion theorem ,Numerical Analysis (math.NA) ,Computational Mathematics ,symbols.namesake ,Computational Theory and Mathematics ,Fourier analysis ,Discrete Fourier series ,FOS: Mathematics ,symbols ,Mathematics - Numerical Analysis ,Fourier series ,Analysis ,Mathematics ,Sine and cosine transforms ,Fourier transform on finite groups - Abstract
An effective means to approximate an analytic, nonperiodic function on a bounded interval is by using a Fourier series on a larger domain. When constructed appropriately, this so-called Fourier extension is known to converge geometrically fast in the truncation parameter. Unfortunately, computing a Fourier extension requires solving an ill-conditioned linear system, and hence one might expect such rapid convergence to be destroyed when carrying out computations in finite precision. The purpose of this paper is to show that this is not the case. Specifically, we show that Fourier extensions are actually numerically stable when implemented in finite arithmetic, and achieve a convergence rate that is at least superalgebraic. Thus, in this instance, ill-conditioning of the linear system does not prohibit a good approximation. In the second part of this paper we consider the issue of computing Fourier extensions from equispaced data. A result of Platte et al. (SIAM Rev. 53(2):308---318, 2011) states that no method for this problem can be both numerically stable and exponentially convergent. We explain how Fourier extensions relate to this theoretical barrier, and demonstrate that they are particularly well suited for this problem: namely, they obtain at least superalgebraic convergence in a numerically stable manner.
- Published
- 2014
36. Growth estimates for arbitrary sequences of multiple rectangular Fourier sums
- Author
-
N. Yu. Antonov
- Subjects
Uses of trigonometry ,Fourier inversion theorem ,Mathematical analysis ,Trigonometric polynomial ,Trigonometric series ,symbols.namesake ,Mathematics (miscellaneous) ,Fourier analysis ,Discrete Fourier series ,Conjugate Fourier series ,symbols ,Computer Science::Symbolic Computation ,Fourier series ,Mathematics - Abstract
Growth estimates are obtained on a set of full measure for arbitrary sequences of rectangular partial sums of multiple trigonometric Fourier series.
- Published
- 2014
37. The Fourier approximation of smooth but non-periodic functions from unevenly spaced data
- Author
-
Mark Lyon and J. Picard
- Subjects
Discrete-time Fourier transform ,Non-uniform discrete Fourier transform ,Applied Mathematics ,Fourier inversion theorem ,Mathematical analysis ,Discrete Fourier transform ,Computational Mathematics ,symbols.namesake ,Fourier analysis ,Discrete Fourier series ,symbols ,Applied mathematics ,Pseudo-spectral method ,Fourier series ,Mathematics - Abstract
We develop an algorithm to extend, to the nonequispaced case, a recently-introduced fast algorithm for constructing spectrally-accurate Fourier approximations of smooth, but nonperiodic, data. Fast Fourier continuation algorithms, which allow for the Fourier approximation to be periodic in an extended domain, are combined with the underlying ideas behind nonequispaced fast Fourier transform (NFFT) algorithms. The result is a method which allows for the fast and accurate approximation of unevenly sampled nonperiodic multivariate data by Fourier series. A particular contribution of the proposed method is that its formulation avoids the difficulties related to the conditioning of the linear systems that must be solved in order to construct a Fourier continuation. The efficiency, essentially equivalent to that of an NFFT, and accuracy of the algorithm is shown through a number of numerical examples. Numerical results demonstrate the spectral rate of convergence of this method for sufficiently smooth functions. The accuracy, for sufficiently large data sets, is shown to be improved by several orders of magnitudes over previously published techniques for scattered data interpolation.
- Published
- 2014
38. Convergence in measure of logarithmic means of multiple Fourier series
- Author
-
Larry Gogoladze and Ushangi Goginava
- Subjects
Control and Optimization ,Convergence in measure ,Applied Mathematics ,Discrete Fourier series ,Fourier inversion theorem ,Mathematical analysis ,Conjugate Fourier series ,Function series ,Half range Fourier series ,Measure (mathematics) ,Fourier series ,Analysis ,Mathematics - Abstract
The maximal Orlicz space such that the mixed logarithmic means of rectangular partial sums of multiple Fourier series for the functions from this space converge in measure is found.
- Published
- 2014
39. Determination of order of magnitude of multiple Fourier coefficients of functions of bounded ϕ-variation having lacunary Fourier series using Jensen's inequality
- Author
-
Bhikha Lila Ghodadra
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Fourier inversion theorem ,Mathematical analysis ,Lebesgue integration ,symbols.namesake ,Fourier analysis ,Discrete Fourier series ,Bounded function ,symbols ,Lacunary function ,Fourier series ,Jensen's inequality ,Mathematics - Abstract
For a Lebesgue integrable complex-valued function f defined over the m -dimensional torus Tm := [0,2π)m , let f (n) denote the Fourier coefficient of f , where n = (n(1), . . . ,n(m)) ∈ Zm . Recently, in one of our papers [to appear in Mathematical Inequalities & Applications], we have defined the notion of bounded φ -variation for a complex-valued function on a rectangle [a1,b1 ]× . . .× [am,bm ] and studied the order of magnitude of Fourier coefficients of such functions on [0,2π]m . In this paper, the order of magnitude of Fourier coefficients of a function of bounded φ -variation from [0,2π]m to C and having lacunary Fourier series with certain gaps is studied and a generalization of our earlier result (Theorem in [Acta Sci. Math. (Szeged), 78, (2012), 97–109]) is proved. Interestingly, the Jensen’s inequality for integrals is used to prove the main result. Mathematics subject classification (2010): 42B05, 26B30, 26D15.
- Published
- 2014
40. On Fourier transformation of a class of entire functions
- Author
-
Marat Il'darovich Musin and Il'dar Khamitovich Musin
- Subjects
symbols.namesake ,Class (set theory) ,Fourier transform ,Fourier analysis ,Discrete-time Fourier transform ,General Mathematics ,Entire function ,Discrete Fourier series ,Mathematical analysis ,Fourier inversion theorem ,symbols ,Fourier series ,Mathematics - Published
- 2014
41. Odd extension for fourier approximation of nonperiodic functions
- Author
-
R. V. Golovanov, N. N. Kalitkin, and K. I. Lutskiy
- Subjects
Discrete-time Fourier transform ,Mathematical analysis ,Fourier inversion theorem ,Discrete Fourier transform ,Computational Mathematics ,symbols.namesake ,Fourier transform ,Fourier analysis ,Modeling and Simulation ,Discrete Fourier series ,symbols ,Fourier series ,Sine and cosine transforms ,Mathematics - Abstract
Approximation of functions by Fourier series plays an important role in applied digital signal processing. The method of odd extension of a nonperiodic function, which increases smoothness compared to existing methods, is proposed. The method is shown to result in a considerable acceleration of convergence of the Fourier series for this function. The method is generalized to the function of two variables. For the two-dimensional Fourier approximation, the optimal technique of truncating the coefficient matrix is found. The advantage of the method is illustrated by test calculations.
- Published
- 2013
42. Absolute convergence of fourier series of almost-periodic functions
- Author
-
Yu. Kh. Khasanov
- Subjects
symbols.namesake ,Fourier transform ,Fourier analysis ,General Mathematics ,Discrete Fourier series ,Mathematical analysis ,Function series ,Fourier inversion theorem ,Conjugate Fourier series ,symbols ,Trigonometric polynomial ,Fourier series ,Mathematics - Abstract
We present necessary and sufficient conditions for the absolute convergence of the Fourier series of almost-periodic (in the sense of Besicovitch) functions when the Fourier exponents have limit points at infinity or at zero. The structural properties of the functions are described by the modulus of continuity or the modulus of averaging of high order, depending on the behavior of the Fourier exponents. The case of uniform almost-periodic functions of bounded variation is considered.
- Published
- 2013
43. Convergence rate estimates for 'spherical' partial sums of double Fourier series
- Author
-
V. A. Abilov, M. K. Kerimov, and M. V. Abilov
- Subjects
Computational Mathematics ,symbols.namesake ,Fourier transform ,Series (mathematics) ,Discrete Fourier series ,Fourier inversion theorem ,Conjugate Fourier series ,Function series ,Mathematical analysis ,symbols ,Half range Fourier series ,Fourier series ,Mathematics - Abstract
The convergence of Fourier double series of 2π-periodic functions from the space \(\mathbb{L}_2\) is analyzed. The convergence rate of spherical partial sums of a double Fourier series is estimated for some classes of functions characterized by a generalized modulus of continuity.
- Published
- 2013
44. On convergence of rational Fourier series of functions of bounded variations
- Author
-
Qian Tao and Tan LiHui
- Subjects
Pure mathematics ,General Mathematics ,Fourier inversion theorem ,Function series ,Mathematical analysis ,Parseval's theorem ,symbols.namesake ,Generalized Fourier series ,Fourier analysis ,Discrete Fourier series ,Conjugate Fourier series ,symbols ,Fourier series ,Mathematics - Abstract
In this paper, we extended some classical results of Fourier series to rational Fourier series. We give an estimate of convergence rate of the rational Fourier series of functions of bounded variation and an analogous one for the conjugate rational Fourier series. As its applications, we deduce the Dirichlet-Jordans theorem and W. H. Youngs theorem for rational Fourier series of functions of bounded variation. Finally, we extend these two theorems to harmonic bounded variation.
- Published
- 2013
45. Some classes of functions and Fourier coefficients with respect to general orthonormal systems
- Author
-
V. Tsagareishvili and Larry Gogoladze
- Subjects
Pure mathematics ,Mathematics (miscellaneous) ,Generalized Fourier series ,Normal convergence ,Discrete Fourier series ,Fourier inversion theorem ,Mathematical analysis ,Conjugate Fourier series ,Function series ,Trigonometric polynomial ,Fourier series ,Mathematics - Abstract
The a.e. convergence of an orthogonal series on [0, 1] depends strongly on the coefficients of this series. It is well known that a sufficient condition for the a.e. convergence of such a series is given by the Men’shov-Rademacher theorem. On the other hand, S. Banach proved that good differential properties of a function do not guarantee the a.e. convergence on [0, 1] of the Fourier series of this function with respect to general orthonormal systems (ONSs). In the present study, we find conditions on the functions of an ONS under which the Fourier coefficients of functions of some differential classes satisfy the hypothesis of the Men’shov-Rademacher theorem.
- Published
- 2013
46. On a class of summability methods for multiple Fourier series
- Author
-
Mikhail Ivanovich Dyachenko
- Subjects
symbols.namesake ,Algebra and Number Theory ,Uses of trigonometry ,Fourier analysis ,Discrete Fourier series ,Mathematical analysis ,Fourier sine and cosine series ,Function series ,Conjugate Fourier series ,Fourier inversion theorem ,symbols ,Fourier series ,Mathematics - Abstract
The paper shows that the same properties which hold for the classical -means are preserved for a sufficiently large class of summability methods for multiple Fourier series involving rectangular partial sums. More precisely, Fourier series of continuous functions are uniformly summable by these methods, and Fourier series of functions from the class are summable almost everywhere. Bibliography: 6 titles.
- Published
- 2013
47. Wiener's theorem for periodic at infinity functions with summable weighted Fourier series
- Author
-
Irina Igorevna Strukova
- Subjects
symbols.namesake ,Fourier transform ,Fourier analysis ,Discrete-time Fourier transform ,General Mathematics ,Discrete Fourier series ,Mathematical analysis ,Convergence of Fourier series ,Fourier inversion theorem ,symbols ,Trigonometric polynomial ,Fourier series ,Mathematics - Abstract
In the article we define a Banach algebra of periodic at infinity functions. For this class of functions we introduce the notions of a Fourier series, its absolute convergence, and invertibility. We obtain an analogue of Wiener's theorem on absolutely convergent Fourier series for periodic at infinity functions whose Fourier coefficients are summable with a weight.
- Published
- 2013
48. Improved approximation guarantees for sublinear-time Fourier algorithms
- Author
-
Mark A. Iwen
- Subjects
Signal recovery ,Non-uniform discrete Fourier transform ,Discrete-time Fourier transform ,Applied Mathematics ,Fourier inversion theorem ,Fourier analysis ,Approximation algorithms ,Discrete Fourier transform ,symbols.namesake ,Discrete Fourier series ,symbols ,Fourier series ,Algorithm ,Fast Fourier transforms ,Sine and cosine transforms ,Mathematics - Abstract
In this paper modified variants of the sparse Fourier transform algorithms from Iwen (2010) [32] are presented which improve on the approximation error bounds of the original algorithms. In addition, simple methods for extending the improved sparse Fourier transforms to higher dimensional settings are developed. As a consequence, approximate Fourier transforms are obtained which will identify a near-optimal k-term Fourier series for any given input function, f : [ 0 , 2 π ] D → C , in O ( k 2 ⋅ D 4 ) time (neglecting logarithmic factors). Faster randomized Fourier algorithm variants with runtime complexities that scale linearly in the sparsity parameter k are also presented.
- Published
- 2013
49. Sobolev smoothing of SVD-based Fourier continuations
- Author
-
Mark Lyon
- Subjects
symbols.namesake ,Non-uniform discrete Fourier transform ,Discrete-time Fourier transform ,Fourier analysis ,Discrete Fourier series ,Applied Mathematics ,Fourier inversion theorem ,Mathematical analysis ,symbols ,Fourier series ,Mathematics ,Sine and cosine transforms ,Parseval's theorem - Abstract
A method for calculating Sobolev smoothed Fourier continuations is presented. The method is based on the recently introduced singular value decomposition based Fourier continuation approach. This approach allows for highly accurate Fourier series approximations of non-periodic functions. These super-algebraically convergent approximations can be highly oscillatory in an extended region, contaminating the Fourier coefficients. It is shown that through solving a subsequent least squares problem, a Fourier continuation can be produced which has been dramatically smoothed in that the Fourier coefficients exhibit a prescribed rate of decay as the wave number increases. While the smoothing procedure has no significant negative effect on the accuracy of the Fourier series approximation, in some situations the smoothed continuations can actually yield increased accuracy in the approximation of the function and its derivatives.
- Published
- 2012
- Full Text
- View/download PDF
50. Approximation error in regularized SVD-based Fourier continuations
- Author
-
Mark Lyon
- Subjects
Numerical Analysis ,Non-uniform discrete Fourier transform ,Discrete-time Fourier transform ,Applied Mathematics ,Fourier inversion theorem ,Mathematical analysis ,Computational Mathematics ,symbols.namesake ,Fourier analysis ,Phase correlation ,Discrete Fourier series ,symbols ,Fourier series ,Mathematics ,Sine and cosine transforms - Abstract
We present an analysis of the convergence of recently developed Fourier continuation techniques that incorporates the required truncation of the Singular Value Decomposition (SVD). Through the analysis, the convergence of SVD-based continuations are related to the convergence of any Fourier approximation of similar form, demonstrating the efficiency and accuracy of the numerical method. The analysis determines that the Fourier continuation approximation error can be bounded by a key value that depends only on the parameters of the Fourier continuation and on the points over which it is applied. For any given distribution of points, a finite number of calculations can be performed to obtain this important value. Our numerical computations on evenly spaced points show that as the number of points increases, this quantity converges to a fixed value, allowing for broad conclusions on the convergence of Fourier continuations calculated with truncated SVDs. We conclude that Fourier continuations can obtain super-algebraic or even exponential convergence on evenly spaced points for non-periodic functions until the convergence is limited by a parameter normally chosen near the machine precision accuracy threshold.
- Published
- 2012
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