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We extend to a large class of noncompact harmonic manifolds the inversion formulas for the Radon transform on horospheres in hyperbolic spaces or Damek–Ricci spaces. Horospheres are defined here as level hypersurfaces of Busemann functions. The proof uses harmonic analysis on the manifolds considered, developed in a recent paper by Biswas, Knieper and Peyerimhoff; we also give a concise proof of their Fourier inversion theorem for harmonic manifolds.

The Fourier transform is defined not only for 1D signals but for functions of arbitrary dimension. Thus, 2D images are nothing special from a mathematical point of view.

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.

Harmonic analysis plays an essential role in understanding a host of engineering, mathematical, and scientific ideas. In Harmonic Analysis and Applications, the analysis and synthesis of functions in terms of harmonics is presented in such a way as to demonstrate the vitality, power, elegance, usefulness, and the intricacy and simplicity of the subject. This book is about classical harmonic analysis - a textbook suitable for students, and an essay and general reference suitable for mathematicians, physicists, and others who use harmonic analysis.Throughout the book, material is provided for an upper level undergraduate course in harmonic analysis and some of its applications. In addition, the advanced material in Harmonic Analysis and Applications is well-suited for graduate courses. The course is outlined in Prologue I. This course material is excellent, not only for students, but also for scientists, mathematicians, and engineers as a general reference. Chapter 1 covers the Fourier analysis of integrable and square integrable (finite energy) functions on R. Chapter 2 of the text covers distribution theory, emphasizing the theory's useful vantage point for dealing with problems and general concepts from engineering, physics, and mathematics. Chapter 3 deals with Fourier series, including the Fourier analysis of finite and infinite sequences, as well as functions defined on finite intervals. The mathematical presentation, insightful perspectives, and numerous well-chosen examples and exercises in Harmonic Analysis and Applications make this book well worth having in your collection.

The Banach Gelfand Triple ( S 0 , L 2 , S 0 ′ ) ( R d ) consists of S 0 ( R d ) , ∥ · ∥ S 0 , a very specific Segal algebra as algebra of test functions, the Hilbert space L 2 ( R d ) , ∥ · ∥ 2 and the dual space S 0 ′ ( R d ) , whose elements are also called “mild distributions”. Together they provide a universal tool for Fourier Analysis in its many manifestations. It is indispensable for a proper formulation of Gabor Analysis, but also useful for a distributional description of the classical (generalized) Fourier transform (with Plancherel’s Theorem and the Fourier Inversion Theorem as core statements) or the foundations of Abstract Harmonic Analysis, as it is not difficult to formulate this theory in the context of locally compact Abelian (LCA) groups. A new approach presented recently allows to introduce S 0 ( R d ) , ∥ · ∥ S 0 and hence ( S 0 ′ ( R d ) , ∥ · ∥ S 0 ′ ) , the space of “mild distributions”, without the use of the Lebesgue integral or the theory of tempered distributions. The present notes will describe an alternative, even more elementary approach to the same objects, based on the idea of completion (in an appropriate sense). By drawing the analogy to the real number system, viewed as infinite decimals, we hope that this approach is also more interesting for engineers. Of course it is very much inspired by the Lighthill approach to the theory of tempered distributions. The main topic of this article is thus an outline of the sequential approach in this concrete setting and the clarification of the fact that it is just another way of describing the Banach Gelfand Triple. The objects of the extended domain for the Short-Time Fourier Transform are (equivalence classes) of so-called mild Cauchy sequences (in short ECmiCS). Representatives are sequences of bounded, continuous functions, which correspond in a natural way to mild distributions as introduced in earlier papers via duality theory. Our key result shows how standard functional analytic arguments combined with concrete properties of the Segal algebra S 0 ( R d ) , ∥ · ∥ S 0 can be used to establish this natural identification.

We demonstrate that the Fourier partial sums of a Lipschitz continuous function on the two-dimensional special unitary group converge uniformly to the function.

In this paper for functions on locally compact Vilenkin groups, we prove an analogue of one classical Titchmarsh theorem on the image under the Fourier transform of a set of functions satisfying the Lipschitz condition in L2.

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 .

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.

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.

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.

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.

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.

In this paper, for functions on the group Q p , we prove an analogue of the classical Titchmarsh theorem on description of the image under the Fourier transform of a set of functions satisfying the Lipschitz condition in L 2.

We prove analogues of the Bombieri–Vinogradov Theorem and the Barban–Davenport–Halberstam Theorem on primes in arithmetic progressions for Fourier coefficients of cusp forms.

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.

In this paper we establish the real Paley–Wiener theorems for the quaternion Fourier transform on for quaternion-valued Schwartz functions and -functions, which generalizes the recent results of real Paley–Wiener theorems for scalar- and quaternion-valued -functions.

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 $\mathcal{S}_o$ .

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.

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.

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.

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.

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.

In this paper, a general summability method of multi-dimensional Fourier transforms, the so-called [Formula: see text]-summability, is investigated. It is shown that if [Formula: see text] is in a Herz space, then the summability means [Formula: see text] of a function [Formula: see text] converge to [Formula: see text] at each modified Lebesgue point, whenever [Formula: see text] and [Formula: see text] is in a cone. The same holds for Fourier series. Some special cases of the [Formula: see text]-summation are considered, such as the Weierstrass, Abel, Picard, Bessel, Fejér, Cesàro, de la Vallée-Poussin, Rogosinski and Riesz summations.

Given a function \(f : {\{0,1\}}^n\rightarrow \mathbb {R}\), its Fourier Entropy is defined to be \(-\sum _S {\widehat{f}}^2(S) \log {\widehat{f}}^2(S)\), where \(\hat{f}\) denotes the Fourier transform of f. This quantity arises in a number of applications, especially in the study of Boolean functions. An outstanding open question is a conjecture of Friedgut and Kalai (1996), called the Fourier Entropy Influence (FEI) Conjecture, asserting that the Fourier Entropy of any Boolean function f is bounded above, up to a constant factor, by the total influence (= average sensitivity) of f.

We establish a Fourier inversion theorem for general connected, simply connected nilpotent Lie groups $$G= \hbox {exp}({\mathfrak {g}})$$ by showing that operator fields defined on suitable sub-manifolds of $${\mathfrak {g}}^*$$ are images of Schwartz functions under the Fourier transform. As an application of this result, we provide a complete characterisation of a large class of invariant prime closed two-sided ideals of $$L^1(G)$$ as kernels of sets of irreducible representations of G.

It is proved that a continuous function f on \(\mathbb {R}^n\) is negative definite if and only if it is polynomially bounded and satisfies the inequality \(\mathbb {E} f(X-Y)\le \mathbb {E} f(X+Y)\) for all i.i.d. random vectors X and Y in \(\mathbb {R}^n\). The proof uses Fourier transforms of tempered distributions. The “only if” part has been proved earlier by Lifshits et al. (A probabilistic inequality related to negative definite functions. Progress in probability, vol. 66 (Springer, Basel, 2013), pp. 73–80).

By using the convolution theorem of the Fourier transform (Faltung theorem), the following integral involving the Macdonald function is calculated, ∫ − ∞ ∞ | t ′ | α + 2 n K α ( a | t ′ | ) | t − t ′ | β + 2 m K β ( a | t − t ′ | ) d t ′ . As a consistency test of the result obtained, setting the parameters α, β, m, n, t and a to particular values, some integrals reported in the literature are recovered. It turns out that the calculation method of integrals via the convolution theorem is useful for calculating other infinite integrals involving the Macdonald function.

We investigate the pointwise and uniform convergence of the symmetric rectangular partial (also called Dirichlet) integrals of the double Fourier integral of a function that is Lebesgue integrable and of bounded variation over ℝ2. Our theorem is a two-dimensional extension of a theorem of Móricz (see Theorem 3 in [10]) concerning the single Fourier integrals, which is more general than the two-dimensional extension given by Móricz himself (see Theorem 3 in [11]).

In computability theory a variety of combinatorial systems are encountered (word problems, production systems) that exhibit undecidability properties. Here we seek such structures in the realm of Analysis, more specifically in the area of Fourier Analysis. The starting point is that sufficiently strongly convergent Fourier series give rise to predicates in the sense of first order predicate calculus by associating to any s-ary Fourier series the predicate "the Fourier coefficient with index (n(1), ... , n(s)) is non-zero". We introduce production systems, viewed as counterparts of the combinatorial ones, that generate all recursively enumerable predicates in this way using as tools only elementary operations and functions from classical Analysis. The problem arises how simple such a system may be. It turns out that there is a connection between this question and an as yet unproved conjecture by R. Bilchi. This is discussed in the second half of the paper. (C) 2016 Elsevier B.V. All rights reserved.

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.

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.

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).

We provide a variant of Hytonen’s embedding theorem, which allows us to extend and unify several sufficient conditions for a function to be a Fourier multiplier on the real Hardy spaces.

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.

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.

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.

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.”

Suppose that 1

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 .

We establish a general sufficient condition for the possibility of representation of functions $$ f\left( \max \left\{\left|{x}_1\right|,\left|{x}_2\right|\right\}\right) $$ in the form of absolutely convergent double Fourier integrals and study the possibility of its application to various problems of summability of double Fourier series, in particular, by using the Marcinkiewicz–Riesz method.

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.

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.

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

Reconstruction of piecewise smooth functions from non-uniform Fourier data arises in sensing applications such as magnetic resonance imaging (MRI). This paper presents a new method that uses edge information to recover the Fourier transform of a piecewise smooth function from data that is sparsely sampled at high frequencies. The approximation is based on a combination of polynomials multiplied by complex exponentials. We obtain super-algebraic convergence rates for a large class of functions with one jump discontinuity, and geometric convergence rates for functions that decay exponentially fast in the physical domain when the derivatives satisfy a certain bound. Exponential convergence is also proved for piecewise analytic functions of compact support. Our method can also be used to improve initial jump location estimates, which are calculated from the available Fourier data, through an iterative process. Finally, if the Fourier transform is approximated at integer values, then the IFFT can be used to reconstruct the underlying function. Post-processing techniques, such as spectral reprojection, can then be used to reduce Gibbs oscillations.

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.

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.