776 results on '"Fourier inversion theorem"'
Search Results
2. Revisiting the Fourier transform on the Heisenberg group
- Author
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Sundaram Thangavelu and R. Lakshmi Lavanya
- Subjects
Mellin transform ,Pure mathematics ,General Mathematics ,Mathematical analysis ,Fourier inversion theorem ,Heisenberg group ,Fractional Fourier transform ,Parseval's theorem ,symbols.namesake ,Discrete Fourier transform (general) ,Fourier transform ,Fourier-Weyl transform ,Hartley transform ,symbols ,Mathematics::Metric Geometry ,Schwartz class ,Mathematics ,Fourier transform on finite groups ,Heat kernel - Abstract
A recent theorem of S. Alesker, S. Artstein-Avidan and V. Milman characterises the Fourier transform on R-n as essentially the only transform on the space of tempered distributions which interchanges convolutions and pointwise products. In this note we study the image of the Schwartz space on the Heisenberg group under the Fourier transform and obtain a similar characterisation for the Fourier transform on the Heisenberg group.
- Published
- 2021
3. 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
4. Some extremal problems for the Fourier transform on the hyperboloid
- Author
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V. I. Ivanov, D. V. Gorbachev, and O. I. Smirnov
- Subjects
Mathematics::Combinatorics ,Discrete-time Fourier transform ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Fourier inversion theorem ,010103 numerical & computational mathematics ,Space (mathematics) ,01 natural sciences ,Fractional Fourier transform ,Discrete Fourier transform (general) ,symbols.namesake ,Fourier transform ,symbols ,Mathematics::Metric Geometry ,0101 mathematics ,Hyperboloid ,Mathematics ,Fourier transform on finite groups - Abstract
We give the solution of the Turan, Fejer, Delsarte, Logan, and Bohman extremal problems for the Fourier transform on the hyperboloid ℍ d or Lobachevsky space. We apply the averaging function method over the sphere and the solution of these problems for the Jacobi transform on the half-line.
- Published
- 2017
5. Fourier transformation of O(p,q)-invariant distributions. Fundamental solutions of ultra-hyperbolic operators
- Author
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Norbert Ortner and Peter Wagner
- Subjects
Applied Mathematics ,010102 general mathematics ,Fourier inversion theorem ,Mathematical analysis ,010103 numerical & computational mathematics ,01 natural sciences ,Fourier integral operator ,Fractional Fourier transform ,Multiplier (Fourier analysis) ,symbols.namesake ,Fourier transform ,Homogeneous ,Fundamental solution ,symbols ,0101 mathematics ,Analysis ,Mathematics - Abstract
In this study, we present some formulae for the Fourier transform of O ( p , q ) -invariant temperate distributions. The formulae are used to calculate fundamental solutions of homogeneous and non-homogeneous ultra-hyperbolic operators.
- Published
- 2017
6. 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
7. Fourier transform and quasi-analytic classes of functions of bounded type on tubular domains
- Author
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F. A. Shamoyan
- Subjects
Hankel transform ,Applied Mathematics ,Fourier inversion theorem ,Mathematical analysis ,Fractional Fourier transform ,symbols.namesake ,Discrete Fourier transform (general) ,Fourier transform ,Fourier analysis ,symbols ,Analysis ,Sine and cosine transforms ,Mathematics ,Fourier transform on finite groups - Abstract
A condition for a function of bounded type to belong to the Hardy class H 1 in terms of the Fourier transform of the boundary values of this function on R n is found. Applications of the obtained result to the theories of Hardy classes and of quasi-analytic classes of functions are given.
- Published
- 2017
8. The Fourier Transform of a Function of Bounded Variation: Symmetry and Asymmetry
- Author
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Elijah Liflyand
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Fourier inversion theorem ,Short-time Fourier transform ,010103 numerical & computational mathematics ,01 natural sciences ,Fractional Fourier transform ,symbols.namesake ,Discrete Fourier transform (general) ,Fourier transform ,Discrete sine transform ,symbols ,0101 mathematics ,Fourier series ,Analysis ,Sine and cosine transforms ,Mathematics - Abstract
New relations between the Fourier transform of a function of bounded variation and the Hilbert transform of its derivative are revealed. The main result of the paper is an asymptotic formula for the cosine Fourier transform. Such relations have previously been known only for the sine Fourier transform. For this, not only a different space is considered but also a new way of proving such theorems is applied. Interrelations of various function spaces are studied in this context. The obtained results are used for obtaining completely new results on the integrability of trigonometric series.
- Published
- 2017
9. Modular interpolation and modular estimates of the Fourier transform and related operators
- Author
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Kwok Pun Victor Ho
- Subjects
Non-uniform discrete Fourier transform ,General Mathematics ,010102 general mathematics ,Fourier inversion theorem ,01 natural sciences ,Fractional Fourier transform ,010101 applied mathematics ,Algebra ,symbols.namesake ,Discrete Fourier transform (general) ,Hartley transform ,symbols ,0101 mathematics ,Trigonometric interpolation ,Interpolation ,Mathematics ,Fourier transform on finite groups - Published
- 2017
10. Pseudo-differential operators involving Fractional Fourier cosine (sine) transform
- Author
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Manoj Kumar Singh and Akhilesh Prasad
- Subjects
symbols.namesake ,Discrete Fourier transform (general) ,Fourier transform ,Discrete sine transform ,General Mathematics ,Fourier sine and cosine series ,Fourier inversion theorem ,Mathematical analysis ,symbols ,Fourier series ,Fractional Fourier transform ,Sine and cosine transforms ,Mathematics - Abstract
A brief introduction to the fractional Fourier cosine transform as well as fractional Fourier sine transform and their basic properties are given. Fractional Fourier cosine (fractional Fourier sine) transform of tempered distributions is studied. Pseudo-differential operators involving these transformations are investigated and discussed the continuity on certain spaces $\mathcal{S}_e $ and $\mathcal{S}_o$ .
- Published
- 2017
11. Time-invariant radon transform by generalized Fourier slice theorem
- Author
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Mauricio D. Sacchi and Ali Gholami
- Subjects
Control and Optimization ,Radon transform ,Mathematical analysis ,Fourier inversion theorem ,020206 networking & telecommunications ,02 engineering and technology ,010502 geochemistry & geophysics ,01 natural sciences ,Fractional Fourier transform ,Parseval's theorem ,symbols.namesake ,Discrete Fourier transform (general) ,Fourier transform ,Modeling and Simulation ,Projection-slice theorem ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,Discrete Mathematics and Combinatorics ,Pharmacology (medical) ,Convolution theorem ,Analysis ,0105 earth and related environmental sciences ,Mathematics - Abstract
Time-invariant Radon transforms play an important role in many fields of imaging sciences, whereby a function is transformed linearly by integrating it along specific paths, e.g. straight lines, parabolas, etc. In the case of linear Radon transform, the Fourier slice theorem establishes a simple analytic relationship between the 2-D Fourier representation of the function and the 1-D Fourier representation of its Radon transform. However, the theorem can not be utilized for computing the Radon integral along paths other than straight lines. We generalize the Fourier slice theorem to make it applicable to general time-invariant Radon transforms. Specifically, we derive an analytic expression that connects the 1-D Fourier coefficients of the function to the 2-D Fourier coefficients of its general Radon transform. For discrete data, the model coefficients are defined over the data coefficients on non-Cartesian points. It is shown numerically that a simple linear interpolation provide satisfactory results and in this case implementations of both the inverse operator and its adjoint are fast in the sense that they run in \begin{document}$O(N \;\text{log}\; N)$\end{document} flops, where \begin{document}$N$\end{document} is the maximum number of samples in the data space or model space. These two canonical operators are utilized for efficient implementation of the sparse Radon transform via the split Bregman iterative method. We provide numerical examples showing high-performance of this method for noise attenuation and wavefield separation in seismic data.
- Published
- 2017
12. Supersymmetric Resolvent-Based Fourier Transform
- Author
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Seiichi Kuwata
- Subjects
010102 general mathematics ,Fourier inversion theorem ,Resolvent formalism ,01 natural sciences ,Fractional Fourier transform ,Parseval's theorem ,symbols.namesake ,Fourier transform ,Quantum mechanics ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Fourier series ,Mathematical physics ,Mathematics ,Resolvent ,Fourier transform on finite groups - Abstract
We calculate in a numerically friendly way the Fourier transform of a non-integrable function, such as , by replacing F with R-1FR, where R represents the resolvent for harmonic oscillator Hamiltonian. As contrasted with the non-analyticity of at in the case of a simple replacement of F by , where and represent the momentum and position operators, respectively, the turns out to be an entire function. In calculating the resolvent kernel, the sampling theorem is of great use. The resolvent based Fourier transform can be made supersymmetric (SUSY), which not only makes manifest the usefulness of the even-odd decomposition ofin a more natural way, but also leads to a natural definition of SUSY Fourier transform through the commutativity with the SUSY resolvent.
- Published
- 2017
13. Properties of the distributional finite Fourier transform
- Author
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Richard D. Carmichael
- Subjects
Analytic functions ,distributions ,finite Fourier transform ,Cauchy integral ,Non-uniform discrete Fourier transform ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Fourier inversion theorem ,Short-time Fourier transform ,010103 numerical & computational mathematics ,01 natural sciences ,Fractional Fourier transform ,Discrete Fourier transform (general) ,symbols.namesake ,Fourier transform ,symbols ,0101 mathematics ,Harmonic wavelet transform ,Fourier transform on finite groups ,Mathematics - Abstract
The analytic functions in tubes which obtain the distributional finite Fourier transform as boundary value are shown to have a strong boundedness property and to be recoverable as a Fourier-Laplace transform, a distributional finite Fourier transform, and as a Cauchy integral of a distribution associated with the boundary value.
- Published
- 2016
14. Quaternionic one-dimensional fractional Fourier transform
- Author
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Rajakumar Roopkumar
- Subjects
Pure mathematics ,Discrete-time Fourier transform ,010102 general mathematics ,Fourier inversion theorem ,020206 networking & telecommunications ,02 engineering and technology ,01 natural sciences ,Atomic and Molecular Physics, and Optics ,Fractional Fourier transform ,Electronic, Optical and Magnetic Materials ,Convolution ,Parseval's theorem ,symbols.namesake ,Discrete Fourier transform (general) ,Fourier transform ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,0101 mathematics ,Electrical and Electronic Engineering ,Fourier series ,Mathematics - Abstract
In this paper, we introduce quaternionic fractional Fourier transform of integrable (and square integrable) functions on ℝ and prove that it is satisfying all the expected properties like linearity, inversion formula, Parseval's formula, convolution theorem and product theorem.
- Published
- 2016
15. Theoretical Elements in Fourier Analysis of q-Gaussian Functions
- Author
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Gilson A. Giraldi and Paulo Sergio Rodrigues
- Subjects
Mathematical optimization ,Discrete-time Fourier transform ,Computer science ,Fourier inversion theorem ,Short-time Fourier transform ,02 engineering and technology ,01 natural sciences ,Discrete Fourier transform ,Fractional Fourier transform ,010305 fluids & plasmas ,Gaussian filter ,symbols.namesake ,Fourier transform ,Fourier analysis ,0103 physical sciences ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,Applied mathematics ,020201 artificial intelligence & image processing - Abstract
There is a consensus in signal processing that the Gaussian kernel and its partial derivatives enable the development of robust algorithms for feature detection. Fourier analysis and convolution theory have a central role in such development. In this paper, we collect theoretical elements to follow this avenue but using the q-Gaussian kernel that is a nonextensive generalization of the Gaussian one. Firstly, we review the one-dimensional q-Gaussian and its Fourier transform. Then, we consider the two-dimensional q-Gaussian and we highlight the issues behind its analytical Fourier transform computation. In the computational experiments, we analyze the q-Gaussian kernel in the space and Fourier domains using the concepts of space window, cut-off frequency, and the Heisenberg inequality.
- Published
- 2016
16. ON UNIFORM SAMPLING IN SHIFT-INVARIANT SPACES ASSOCIATED WITH THE FRACTIONAL FOURIER TRANSFORM DOMAIN
- Author
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Sinuk Kang
- Subjects
symbols.namesake ,Discrete Fourier transform (general) ,Fourier transform ,Fourier analysis ,Discrete-time Fourier transform ,Non-uniform discrete Fourier transform ,Mathematical analysis ,Fourier inversion theorem ,symbols ,Fractional Fourier transform ,Mathematics ,Fourier transform on finite groups - Published
- 2016
17. Asymptotics of the Fourier sine transform of a function of bounded variation
- Author
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E. R. Liflyand
- Subjects
Mellin transform ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Fourier inversion theorem ,010103 numerical & computational mathematics ,01 natural sciences ,Bounded mean oscillation ,Fractional Fourier transform ,Discrete Fourier transform (general) ,symbols.namesake ,Fourier transform ,Discrete sine transform ,symbols ,0101 mathematics ,Mathematics ,Sine and cosine transforms - Abstract
For the asymptotic formula for the Fourier sine transform of a function of bounded variation, we find a new proof entirely within the framework of the theory of Hardy spaces, primarily with the use of the Hardy inequality. We show that, for a function of bounded variation whose derivative lies in the Hardy space, every aspect of the behavior of its Fourier transform can somehow be expressed in terms of the Hilbert transform of the derivative.
- Published
- 2016
18. Paley-Wiener Theorems for the Two-Sided Quaternion Fourier Transform
- Author
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Hatem Mejjaoli
- Subjects
Discrete-time Fourier transform ,Applied Mathematics ,010102 general mathematics ,Fourier inversion theorem ,01 natural sciences ,Fractional Fourier transform ,Parseval's theorem ,010101 applied mathematics ,Algebra ,symbols.namesake ,Discrete Fourier transform (general) ,Fourier transform ,Projection-slice theorem ,Hartley transform ,symbols ,0101 mathematics ,Mathematics - Abstract
In this paper we establish new real Paley-Wiener theorems for the two-sided quaternion Fourier transform. Next, we prove the Roe’s theorem in the context of the quaternion-valued functions. Finally we study the tempered distributions with spectral gaps.
- Published
- 2016
19. Behavior of the First Variation of Fourier Transform of a Measure on the Fourier Feynman Transform and Convolution
- Author
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Young Sik Kim
- Subjects
Control and Optimization ,Discrete-time Fourier transform ,010102 general mathematics ,Fourier inversion theorem ,Mathematical analysis ,010103 numerical & computational mathematics ,01 natural sciences ,Fractional Fourier transform ,Computer Science Applications ,Convolution ,symbols.namesake ,Discrete Fourier transform (general) ,Fourier transform ,Signal Processing ,Hartley transform ,symbols ,0101 mathematics ,Analysis ,Mathematics ,Fourier transform on finite groups - Abstract
We investigate the behavior of a first variation of a Fourier transform of a measure of the form : upon the Fourier-Feynman transform and analytic Feynman integral and the convolution, where and {h1, h2,…, hn} is the orthonormal class of elements in H on the abstract Wiener space (H, B, m).
- Published
- 2016
20. Integrability spaces for the Fourier transform of a function of bounded variation
- Author
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Elijah Liflyand
- Subjects
Applied Mathematics ,010102 general mathematics ,Fourier inversion theorem ,Mathematical analysis ,Bounded deformation ,01 natural sciences ,Bounded mean oscillation ,Fractional Fourier transform ,010101 applied mathematics ,symbols.namesake ,Fourier transform ,Bounded function ,symbols ,0101 mathematics ,Fourier series ,Analysis ,Sine and cosine transforms ,Mathematics - Abstract
New relations between the Fourier transform of a function of bounded variation and the Hilbert transform of its derivative are revealed. After various preceding works of the last 25 years where the behavior of the Fourier transform has been considered on specific subspaces of the space of functions of bounded variation, in this paper such problems are considered on the whole space of functions of bounded variation. The widest subspaces of the space of functions of bounded variation are studied for which the cosine and sine Fourier transforms are integrable. The main result of the paper is an asymptotic formula for the sine Fourier transform of an arbitrary locally absolutely continuous function of bounded variation. Interrelations of various function spaces are studied, in particular, the sharpness of Hardy's inequality is established and the inequality itself is strengthened in certain cases. A way to extend the obtained results to the radial case is shown.
- Published
- 2016
21. A generalized convolution theorem for the special affine Fourier transform and its application to filtering
- Author
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Xiyang Zhi, Deyun Wei, and Wei Zhang
- Subjects
Overlap–add method ,Discrete-time Fourier transform ,Mathematical analysis ,Fourier inversion theorem ,020206 networking & telecommunications ,02 engineering and technology ,01 natural sciences ,Atomic and Molecular Physics, and Optics ,Fractional Fourier transform ,Circular convolution ,Electronic, Optical and Magnetic Materials ,010309 optics ,symbols.namesake ,Fourier transform ,0103 physical sciences ,Hartley transform ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,Electrical and Electronic Engineering ,Convolution theorem ,Mathematics - Abstract
The special affine Fourier transform (SAFT), which is a time-shifted and frequency-modulated version of the linear canonical transform (LCT), has been shown to be a powerful tool for signal processing and optics. Many properties for this transform are already known, but an extension of convolution theorem of Fourier transform (FT) is still not having a widely accepted closed form expression. The purpose of this paper is to introduce a new convolution structure for the SAFT that preserves the convolution theorem for the FT, which states that the FT of the convolution of two functions is the product of their Fourier transforms. Moreover, some of well-known results about the convolution theorem in FT domain, fractional Fourier transform (FRFT) domain, LCT domain are shown to be special cases of our achieved results. Last, as an application, utilizing the new convolution theorem, we investigate the multiplicative filter in the SAFT domain. The new convolution structure is easy to implement in the designing of filters.
- Published
- 2016
22. On the amplitude and phase response in the non-abelian Fourier transform
- Author
-
Peter Zizler
- Subjects
Algebra and Number Theory ,Discrete-time Fourier transform ,Adaptive-additive algorithm ,Mathematical analysis ,Fourier inversion theorem ,Short-time Fourier transform ,010103 numerical & computational mathematics ,01 natural sciences ,Fractional Fourier transform ,Discrete Fourier transform (general) ,symbols.namesake ,Fourier transform ,symbols ,0101 mathematics ,Mathematics ,Fourier transform on finite groups - Abstract
In the context of the non-abelian Fourier transform, the natural extension of the amplitude and phase response to a convolution by a given filter mask are shown to be the polar decompositions of the Fourier transform matrices. A specific example regarding amplitude and phase response of a certain filter mask over the symmetric group S_3 is given.
- Published
- 2016
23. An Omega((n log n)/R) Lower Bound for Fourier Transform Computation in the R-Well Conditioned Model
- Author
-
Nir Ailon
- Subjects
Discrete mathematics ,Discrete-time Fourier transform ,Fourier inversion theorem ,Short-time Fourier transform ,020206 networking & telecommunications ,0102 computer and information sciences ,02 engineering and technology ,01 natural sciences ,Fractional Fourier transform ,Theoretical Computer Science ,Parseval's theorem ,Combinatorics ,symbols.namesake ,Fourier transform ,Computational Theory and Mathematics ,Discrete sine transform ,010201 computation theory & mathematics ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,Fourier transform on finite groups ,Mathematics - Abstract
Obtaining a nontrivial (superlinear) lower bound for computation of the Fourier transform in the linear circuit model has been a long-standing open problem for more than 40 years. An early result by Morgenstern from 1973, provides an Ω( n log n ) lower bound for the unnormalized Fourier transform when the constants used in the computation are bounded. The proof uses a potential function related to a determinant. That result does not explain why the normalized Fourier transform (of unit determinant) should be difficult to compute in the same model. Hence, it is not scale insensitive. More recently, Ailon [2013] showed that if only unitary 2-by-2 gates are used, and additionally no extra memory is allowed, then the normalized Fourier transform requires Ω( n log n ) steps. This rather limited result is also sensitive to scaling, but highlights the complexity inherent in the Fourier transform arising from introducing entropy, unlike, say, the identity matrix (which is as complex as the Fourier transform using Morgenstern’s arguments, under proper scaling). This work improves on Ailon [2013] in two ways: First, we eliminate the scaling restriction and provide a lower bound for computing any scaling of the Fourier transform. Second, we allow the computational model to use extra memory. Our restriction is that the composition of all gates up to any point must be a well- conditioned linear transformation. The lower bound is Ω( R −1 n log n ), where R is the uniform condition number. Well-conditioned is a natural requirement for algorithms accurately computing linear transformations on machine architectures of bounded word size. Hence, this result can be seen as a tradeoff between speed and accuracy. The main technical contribution is an extension of matrix entropy used in Ailon [2013] for unitary matrices to a potential function computable for any invertible matrix, using “quasi-entropy” of “quasi-probabilities.”
- Published
- 2016
24. Beurling's theorem for the Bessel–Struve transform
- Author
-
Hind Lahlali, Radouan Daher, and Azzedine Achak
- Subjects
Mellin transform ,Hankel transform ,Fourier inversion theorem ,Mathematical analysis ,Mathematics::Classical Analysis and ODEs ,General Medicine ,Physics::History of Physics ,Fractional Fourier transform ,symbols.namesake ,Discrete Fourier transform (general) ,Fourier transform ,Projection-slice theorem ,symbols ,Two-sided Laplace transform ,Mathematics - Abstract
The Bessel–Struve transform satisfies some uncertainty principles in a similar way to the Euclidean Fourier transform. Beurling's theorem is obtained for the Bessel–Struve transform F B , S α .
- Published
- 2016
25. Direct Inversion of the Three-Dimensional Pseudo-polar Fourier Transform
- Author
-
Yoel Shkolnisky, Amir Averbuch, and Gil Shabat
- Subjects
Discrete-time Fourier transform ,Non-uniform discrete Fourier transform ,Applied Mathematics ,Fourier inversion theorem ,Mathematical analysis ,Short-time Fourier transform ,020206 networking & telecommunications ,02 engineering and technology ,Fractional Fourier transform ,Discrete Fourier transform ,Computational Mathematics ,symbols.namesake ,Fourier transform ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,020201 artificial intelligence & image processing ,Harmonic wavelet transform ,Mathematics - Abstract
The pseudo-polar Fourier transform is a specialized nonequally spaced Fourier transform, which evaluates the Fourier transform on a near-polar grid known as the pseudo-polar grid. The advantage of the pseudo-polar grid over other nonuniform sampling geometries is that the transformation, which samples the Fourier transform on the pseudo-polar grid, can be inverted using a fast and stable algorithm. For other sampling geometries, even if the nonequally spaced Fourier transform can be inverted, the only known algorithms are iterative. The convergence speed of these algorithms and their accuracy are difficult to control, as they depend both on the sampling geometry and on the unknown reconstructed object. In this paper, a direct inversion algorithm for the three-dimensional pseudo-polar Fourier transform is presented. The algorithm is based only on one-dimensional resampling operations and is shown to be significantly faster than existing iterative inversion algorithms.
- Published
- 2016
26. Applications of the Funk–Hecke theorem to smoothing and trace estimates
- Author
-
Hiroki Saito, Neal Bez, and Mitsuru Sugimoto
- Subjects
symbols.namesake ,Fourier transform ,Conjecture ,Discrete-time Fourier transform ,General Mathematics ,Projection-slice theorem ,Fourier inversion theorem ,Mathematical analysis ,symbols ,Fourier series ,Fractional Fourier transform ,Mathematics ,Parseval's theorem - Abstract
For a wide class of Kato-smoothing estimates with radial weights, the Funk–Hecke theorem is used to generate a new expression for the optimal constant in terms of the Fourier transform of the weight, from which several applications are given. For example, we are able to easily establish a unified theorem, assuming natural power-like asymptotic estimates for the Fourier transform of the weight, from which many well-studied smoothing estimates immediately follow, as well as sharpness of the decay and smoothness exponents. Furthermore, observing that the weight has an everywhere positive Fourier transform in many well-studied cases, our approach allows sharper information regarding the optimal constant and extremisers, substantially extending earlier work of Simon. These observations are very closely related to the Mizohata–Takeuchi conjecture regarding the equivalence of weighted L 2 bounds for the Fourier extension operator on the sphere and the uniform boundedness of the X-ray transform of the weight. For radial weights, this has been independently established by Barcelo–Ruiz–Vega and Carbery–Soria; we provide a short alternative proof in three and higher dimensions of this equivalence when the Fourier transform of the weight is positive, with the optimal relationship between constants. Finally, our approach works for the closely connected trace theorems on the sphere where analogous results are given, including the optimal constant and characterisation of extremisers for the inhomogeneous H s ( R d ) → L 2 ( S d − 1 ) trace theorem.
- Published
- 2015
27. Time-frequency localization for the short time Fourier transform
- Author
-
H. Lamouchi and Slim Omri
- Subjects
Discrete-time Fourier transform ,Non-uniform discrete Fourier transform ,Applied Mathematics ,010102 general mathematics ,Fourier inversion theorem ,Mathematical analysis ,Short-time Fourier transform ,01 natural sciences ,Fractional Fourier transform ,Parseval's theorem ,010101 applied mathematics ,Discrete Fourier transform (general) ,symbols.namesake ,Fourier transform ,symbols ,0101 mathematics ,Analysis ,Mathematics - Abstract
We use some estimating of orthogonal projection in a reproducing kernel Hilbert space, to prove a sharp quantitaive form of Shapiro's mean dispersion theroem with generalized dispersion for the short time Fourier transform. Other forms of localization of orthonormal sequences in L2ℝd) notably the umbrella theorem, are also proved for the short time Fourier transform.
- Published
- 2015
28. The Schwartz Space and the Fourier Transform
- Author
-
Dorina Mitrea
- Subjects
Scale space implementation ,symbols.namesake ,Discrete Fourier transform (general) ,Fourier transform ,Schwartz space ,Hartley transform ,Fourier inversion theorem ,Mathematical analysis ,symbols ,Fractional Fourier transform ,Mathematics ,Fourier transform on finite groups - Abstract
This chapter contains material pertaining to the Schwartz space of functions rapidly decaying at infinity and the Fourier transform in such a setting.
- Published
- 2018
29. The generalized integral Fourier transform
- Author
-
M.O. Chetvertak and N.O. Virchenko
- Subjects
Discrete Fourier transform (general) ,symbols.namesake ,Fourier transform ,Non-uniform discrete Fourier transform ,Mathematical analysis ,Hartley transform ,Fourier inversion theorem ,symbols ,Integral transform ,Fractional Fourier transform ,Fourier transform on finite groups ,Mathematics - Published
- 2015
30. Tighter Uncertainty Principles Based on Quaternion Fourier Transform
- Author
-
Yan Yang, Tao Qian, and Pei Dang
- Subjects
Discrete-time Fourier transform ,Applied Mathematics ,010102 general mathematics ,Fourier inversion theorem ,Mathematical analysis ,Short-time Fourier transform ,02 engineering and technology ,01 natural sciences ,Fractional Fourier transform ,Discrete Fourier transform (general) ,symbols.namesake ,Fourier transform ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,020201 artificial intelligence & image processing ,Mathematics::Differential Geometry ,0101 mathematics ,Polar coordinate system ,Harmonic wavelet transform ,Mathematics - Abstract
The quaternion Fourier transform (QFT) and its properties are reviewed in this paper. Under the polar coordinate form for quaternion-valued signals, we strengthen the stronger uncertainty principles in terms of covariance for quaternion-valued signals based on the right-sided quaternion Fourier transform in both the directional and the spatial cases. We also obtain the conditions that give rise to the equal relations of two uncertainty principles. Examples are given to verify the results.
- Published
- 2015
31. uncertainty principles for the Fourier transform with numerical aspect
- Author
-
Fethi Soltani and Akram Nemri
- Subjects
Uncertainty principle ,Applied Mathematics ,010102 general mathematics ,Fourier inversion theorem ,Mathematical analysis ,Conjugate variables ,01 natural sciences ,Fractional Fourier transform ,Discrete Fourier transform ,symbols.namesake ,Fourier transform ,Fourier analysis ,0103 physical sciences ,symbols ,0101 mathematics ,Entropic uncertainty ,010303 astronomy & astrophysics ,Analysis ,Mathematics - Abstract
In this paper, we establish local uncertainty principle for the Fourier transform; and we deduce version of Heisenberg–Pauli–Weyl uncertainty principle. We use also the local uncertainty principle, the partial Fourier integrals and the techniques of Donoho–Stark, we present two uncertainty principles of concentration type in the theory, when . Some numerical applications are given.
- Published
- 2015
32. Some Properties of the Spinor Fourier Transform
- Author
-
Thomas Batard and Tim Raeymaekers
- Subjects
Pure mathematics ,CONVOLUTION ,Discrete-time Fourier transform ,Applied Mathematics ,010102 general mathematics ,Fourier inversion theorem ,02 engineering and technology ,01 natural sciences ,Fractional Fourier transform ,Spinor fourier transform Eigenfunctions Convolution product ,Parseval's theorem ,Algebra ,symbols.namesake ,Discrete Fourier transform (general) ,Mathematics and Statistics ,Fourier transform ,Hartley transform ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,020201 artificial intelligence & image processing ,0101 mathematics ,Fourier transform on finite groups ,Mathematics - Abstract
In this paper, the theory of the spinor Fourier transform introduced in [Batard T, Berthier M, Saint-Jean C, Clifford-Fourier Transform for Color Image Processing, Geometric Algebra Computing for Engineering and Computer Science (E. Bayro-Corrochano and G. Scheuermann Eds.), Springer, London, 2010, pp. 135–161] is further developed. While in the original paper, the transform was determined for vector-valued functions only, it now will be extended to functions taking values in the entire Clifford algebra. Next, two bases are determined under which this Fourier transform is diagonalizable. A main stumbling block for further applications, in particular concerning filter design in the Fourier domain, is the lack of a proper convolution theorem. This problem will be tackled in the final section of this paper.
- Published
- 2015
33. On multiple Fourier coefficients of a function of the generalized Wiener class
- Author
-
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
34. Cowling–Price’s and Hardy’s uncertainty Principles for the generalized Fourier transform associated to a Cherednik type operator on the real line
- Author
-
Hatem Mejjaoli
- Subjects
Pure mathematics ,General Mathematics ,Fourier inversion theorem ,Fractional Fourier transform ,Discrete Fourier transform ,Parseval's theorem ,symbols.namesake ,Fourier transform ,Projection-slice theorem ,Hartley transform ,symbols ,Mathematics::Representation Theory ,Real line ,Mathematics - Abstract
In this paper, we prove the Hardy’s and Cowling–Price’s uncertainty principles for the generalized Fourier transform associated to a Cherednik type operator on the real line.
- Published
- 2015
35. Pitt's inequality and the uncertainty principle associated with the quaternion Fourier transform
- Author
-
Ming-Sheng Liu, Li-Ping Chen, and Kit Ian Kou
- Subjects
Applied Mathematics ,Fourier inversion theorem ,Mathematical analysis ,Short-time Fourier transform ,Fractional Fourier transform ,Parseval's theorem ,symbols.namesake ,Fourier transform ,Fourier analysis ,Hartley transform ,symbols ,Mathematics::Differential Geometry ,Analysis ,Mathematics ,Fourier transform on finite groups - Abstract
The quaternion Fourier transform – a generalized form of the classical Fourier transform – has been shown to be a powerful analyzing tool in image and signal processing. This paper investigates Pitt's inequality and uncertainty principle associated with the two-sided quaternion Fourier transform. It is shown that by applying the symmetric form f = f 1 + i f 2 + f 3 j + i f 4 j of quaternion from Hitzer and the novel module or L p -norm of the quaternion Fourier transform f ˆ , then any nonzero quaternion signal and its quaternion Fourier transform cannot both be highly concentrated. Two part results are provided, one part is Heisenberg–Weyl's uncertainty principle associated with the quaternion Fourier transform. It is formulated by using logarithmic estimates which may be obtained from a sharp of Pitt's inequality; the other part is the uncertainty principle of Donoho and Stark associated with the quaternion Fourier transform.
- Published
- 2015
36. Solution of the Energy Concentration Problem in Reproducing-Kernel Hilbert Space
- Author
-
Ahmed I. Zayed
- Subjects
symbols.namesake ,Discrete Fourier transform (general) ,Fourier transform ,Fourier analysis ,Applied Mathematics ,Mathematical analysis ,Fourier inversion theorem ,Hartley transform ,symbols ,Fractional Fourier transform ,Reproducing kernel Hilbert space ,Sine and cosine transforms ,Mathematics - Abstract
The problem of maximizing the energy of a signal bandlimited to $[-\sigma, \sigma]$ on an interval $[-\tau, \tau]$ in the time domain is one of the important classical problems in signal processing. This problem was solved by a group of mathematicians, Slepian, Landau, and Pollak, at Bell Labs in the 1960s. The solution involved the prolate spheroidal wave functions. More recently, Pei and Ding solved the energy problem for more general transforms than the Fourier transform, such as the fractional Fourier transform and the linear canonical transform. Their solution involved what they called the generalized prolate spheroidal wave functions. The goals of this article are (1) to show that the problem of finding a bandlimited signal with maximum energy concentration in a given interval is a special case of a more general problem in reproducing-kernel Hilbert spaces, (2) to solve the problem in the setting of a Hilbert space and then obtain the solutions for the Fourier, fractional Fourier and linear canonica...
- Published
- 2015
37. On the convolution theorem for the Fourier transform of BV_0 functions
- Author
-
Francisco Javier Mendoza Torres and M. Guadalupe Morales Macías
- Subjects
symbols.namesake ,Discrete Fourier transform (general) ,Fourier transform ,Discrete-time Fourier transform ,Projection-slice theorem ,Fourier inversion theorem ,Mathematical analysis ,Hartley transform ,symbols ,General Medicine ,Fractional Fourier transform ,Convolution ,Mathematics - Published
- 2015
38. On the Reduced Noise Sensitivity of a New Fourier Transformation Algorithm
- Author
-
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
39. Asymptotic behaviour of the quaternion linear canonical transform and the Bochner–Minlos theorem
- Author
-
Kit Ian Kou and João Morais
- Subjects
Lemma (mathematics) ,Quaternion algebra ,Hurwitz quaternion ,Applied Mathematics ,Fourier inversion theorem ,Minlos' theorem ,Fractional Fourier transform ,Algebra ,Computational Mathematics ,symbols.namesake ,Fourier transform ,Projection-slice theorem ,symbols ,Mathematics - Abstract
There have been numerous proposals in the literature to generalize the classical Fourier transform by making use of the Hamiltonian quaternion algebra. The present paper reviews the quaternion linear canonical transform (QLCT) which is a generalization of the quaternion Fourier transform and it studies a number of its properties. In the first part of this paper, we establish a generalized Riemann-Lebesgue lemma for the (right-sided) QLCT. This lemma prescribes the asymptotic behaviour of the QLCT extending and refining the classical Riemann-Lebesgue lemma for the Fourier transform of 2D quaternion signals. We then introduce the QLCT of a probability measure, and we study some of its basic properties such as linearity, reconstruction formula, continuity, boundedness, and positivity. Finally, we extend the classical Bochner-Minlos theorem to the QLCT setting showing the applicability of our approach.
- Published
- 2014
40. 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
41. Appraisal problem in the 3D least squares Fourier seismic data reconstruction
- Author
-
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
42. CONVOLUTION THEOREMS FOR CLIFFORD FOURIER TRANSFORM AND PROPERTIES
- Author
-
Ryuichi Ashino, Rémi Vaillancourt, and Mawardi Bahri
- Subjects
Discrete-time Fourier transform ,lcsh:Mathematics ,Fourier inversion theorem ,Clifford convolution, Clifford algebra, Clifford Fourier transform ,Clifford analysis ,lcsh:QA1-939 ,Fractional Fourier transform ,Convolution ,Algebra ,Discrete Fourier transform (general) ,symbols.namesake ,Fourier transform ,symbols ,Convolution theorem ,Mathematics - Abstract
The non-commutativity of the Clifford multiplication gives different aspects from the classical Fourier analysis.We establish main properties of convolution theorems for the Clifford Fourier transform. Some properties of these generalized convolutionsare extensions of the corresponding convolution theorems of the classical Fourier transform.DOI : http://dx.doi.org/10.22342/jims.20.2.143.125-140
- Published
- 2014
43. Approximations to the Solution of R-L Space Fractional Heat Equation in Terms of Kummers Hyper Geometric Functions by Using Fourier Transform Method: A New Approach
- Author
-
A.P. Bhadane and S.D. Manjarekar
- Subjects
symbols.namesake ,Scale space implementation ,Discrete Fourier transform (general) ,Fourier transform ,Fourier analysis ,Fourier inversion theorem ,Hartley transform ,Mathematical analysis ,symbols ,Fractional Fourier transform ,Fourier transform on finite groups ,Mathematics - Abstract
The purpose of this paper is to give applications of Fourier Transform to solve the Riemann – Liouville Space Fractional Heat equation by using Fourier Transform Method approximated by Kummers Hyper geometric functions.
- Published
- 2016
44. 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
45. Relationship between the cayley-dickson fourier transform and the hartley transform of multidimensional real signals
- Author
-
Kajetana M. Snopek
- Subjects
Pure mathematics ,Non-uniform discrete Fourier transform ,020208 electrical & electronic engineering ,Fourier inversion theorem ,02 engineering and technology ,Fractional Fourier transform ,Discrete Hartley transform ,symbols.namesake ,Discrete Fourier transform (general) ,Hartley transform ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,020201 artificial intelligence & image processing ,Astrophysics::Earth and Planetary Astrophysics ,Harmonic wavelet transform ,Fourier transform on finite groups ,Mathematics - Abstract
The paper shows relations between two different frequency representations of n-dimensional signals (n = 1, 2, 3): the Cayley-Dickson Fourier transform and the Hartley transform. The formulas relating the n-D complex Fourier transform and the n-D Hartley transform are presented. New formulas relating the Quaternion and Octonion Fourier transforms and respectively the 2-D and 3-D Hartley transforms are developed. The paper is illustrated with examples of Hartley transforms of 1-D, 2-D and 3-D Gaussian signals.
- Published
- 2017
46. 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
47. On the fractional Fourier and continuous fractional wave packet transforms of almost periodic functions
- Author
-
Banu Ünalmış Uzun, Işık Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, Işık University, Faculty of Arts and Sciences, Department of Mathematics, and Ünalmış Uzun, Banu
- Subjects
frame ,Discrete-time Fourier transform ,02 engineering and technology ,Fractional fourier transform ,01 natural sciences ,fractional wave packet transform ,010309 optics ,symbols.namesake ,0103 physical sciences ,Hartley transform ,0202 electrical engineering, electronic engineering, information engineering ,Discrete Mathematics and Combinatorics ,Frame ,42A38 ,Mathematics ,Research ,lcsh:Mathematics ,Applied Mathematics ,Fourier inversion theorem ,Mathematical analysis ,Fractional wave packet transform ,020206 networking & telecommunications ,almost periodic function ,fractional Fourier transform ,lcsh:QA1-939 ,Fractional Fourier transform ,Fractional calculus ,Periodic function ,Fourier transform ,Fourier analysis ,44A20 ,symbols ,Almost periodic function ,42A05 ,Analysis - Abstract
PubMed ID: 28680229 We state the fractional Fourier transform and the continuous fractional wave packet transform as ways for analyzing persistent signals such as almost periodic functions and strong limit power signals. We construct frame decompositions for almost periodic functions using these two transforms. Also a norm equality of this signal is given using the continuous fractional wave packet transform. Publisher's Version
- Published
- 2017
48. Note on a non standard eigenfunction of the planar Fourier transform
- Author
-
Flavia Lanzara and Vladimir Maz'ya
- Subjects
Statistics and Probability ,Discrete-time Fourier transform ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Fourier optics ,Mathematical analysis ,Fourier inversion theorem ,Mathematics::Spectral Theory ,Eigenfunction ,01 natural sciences ,Fractional Fourier transform ,010101 applied mathematics ,Discrete Fourier transform (general) ,symbols.namesake ,Fourier transform ,symbols ,0101 mathematics ,Mathematics ,Fourier transform on finite groups - Abstract
We consider a nontrivial example of distributional eigenfunction of the planar Fourier transform. This eigenfunction is not a tensor product of univariate eigenfunctions. As a consequence, we obtain a formula for multi-dimensional eigenfunctions in dimension 2N.
- Published
- 2017
49. Asymptotic Behavior of the Fourier Transform of a Function of Bounded Variation
- Author
-
Elijah Liflyand
- Subjects
Mellin transform ,symbols.namesake ,Discrete Fourier transform (general) ,Fourier transform ,Discrete-time Fourier transform ,Fourier inversion theorem ,Mathematical analysis ,symbols ,Two-sided Laplace transform ,Fourier series ,Fractional Fourier transform ,Mathematics - Abstract
A recent result on the asymptotic behavior of the sine Fourier transform of an arbitrary locally absolutely continuous function of bounded variation is extended to the case of several variables. For this, the initial one-dimensional result is reconsidered and refined. To even one-dimensional asymptotics and their multidimensional generalizations, a new balance operator is introduced.
- Published
- 2017
50. k-deformed Fourier transform
- Author
-
Antonio Maria Scarfone
- Subjects
Statistics and Probability ,$\kappa$-deformed algebra ,FOS: Physical sciences ,01 natural sciences ,power-law distribution ,010305 fluids & plasmas ,Parseval's theorem ,symbols.namesake ,0103 physical sciences ,Fourier integral transform ,010306 general physics ,Fourier series ,Condensed Matter - Statistical Mechanics ,Fourier transform on finite groups ,Mathematics ,log-periodic oscillations ,Statistical Mechanics (cond-mat.stat-mech) ,Mathematical analysis ,Fourier inversion theorem ,Short-time Fourier transform ,Condensed Matter Physics ,Fractional Fourier transform ,Mathematics::Logic ,Fourier transform ,Fourier analysis ,symbols - Abstract
We present a new formulation of Fourier transform in the picture of the $\kappa$-algebra derived in the framework of the $\kappa$-generalized statistical mechanics. The $\kappa$-Fourier transform is obtained from a $\kappa$-Fourier series recently introduced by us [2013 Entropy {\bf15} 624]. The kernel of this transform, that reduces to the usual exponential phase in the $\kappa\to0$ limit, is composed by a $\kappa$-deformed phase and a damping factor that gives a wavelet-like behavior. We show that the $\kappa$-Fourier transform is isomorph to the standard Fourier transform through a changing of time and frequency variables. Nevertheless, the new formalism is useful to study, according to Fourier analysis, those functions defined in the realm of the $\kappa$-algebra. As a relevant application, we discuss the central limit theorem for the $\kappa$-sum of $n$-iterate statistically independent random variables., Comment: 31 pages, 6 figures, elsart stile
- Published
- 2017
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