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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.

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.

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.

We study convergence almost everywhere of multiple trigonometric Fourier series over cubes defined on the d d -dimensional torus T d \mathbb {T}^d . We provide a new approach which allows us to prove the novel interpolation estimates for the Carleson maximal operators generated by the partial sums of the multiple Fourier series and all its conjugate series. Combining these estimates we show that these operators are bounded from a variant of the Arias-de-Reyna space Q A d Q\!A^d to the weak L 1 L^1 -space on T d \mathbb {T}^d . This implies that the multiple Fourier series of every function f ∈ Q A d f\in Q\!A^d and all its conjugate series converge over cubes almost everywhere. By a close analysis of the space Q A d Q\!A^d we prove that it contains a Lorentz space that strictly contains the Orlicz space L ( log L ) d log ⁡ log ⁡ log L ( T d ) L(\log \,L)^{d} \log \log \log \,L(\mathbb {T}^d) . This yields a significant improvement of a deep theorem proved by Antonov which was the best known result on the convergence of multiple Fourier series over cubes.

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.

We study solutions to convolution equations for functions with discrete support in ℝ n , a special case being functions with support in the integer points. The Fourier transform of a solution can be extended to a holomorphic function in some domains in ℂ n , and we determine possible domains in terms of the properties of the convolution operator.

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.

In this work we give Holder–Besov estimates for periodic Fourier multipliers. We present a class of bounded pseudo-differential operators on periodic Besov spaces with symbols of limited regularity.

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 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).

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.

In this paper, we consider the generalized Poisson’s equation associated with Heckman–Opdam Laplacian operator. We introduce and we study the generalized resolvent function associated to the equation. Next, we characterize the spectrum of the hypergeometric Fourier transform via the generalized resolvent function.

It was proved earlier that, for multiple Fourier series whose coefficients are monotone in each index, the classicalHardy–Littlewood theorem is not valid for p ≤ 2m/(m+1), where m is the dimension of the space. We establish how the theorem must be modified in this case.

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.