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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 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 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.

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.

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.

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.

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.

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 .

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.

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.

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.

In this paper we state new theorems of Hardy–Littlewood type for functions with general monotone Fourier coefficients. Sharpness of stated results is discusses.

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.

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.

We establish the regularity of bilinear Fourier integral operators with bilinear amplitudes in S 1 , 0 m ( n , 2 ) and non-degenerate phase functions, from L p × L q → L r under the assumptions that m ⩽ − ( n − 1 ) ( | 1 p − 1 2 | + | 1 q − 1 2 | ) and 1 p + 1 q = 1 r . This is a bilinear version of the classical theorem of Seeger–Sogge–Stein concerning the L p boundedness of linear Fourier integral operators. Moreover, our result goes beyond the aforementioned theorem in that it also includes the case of quasi-Banach target spaces.

The nonlinear Fourier basis has shown its advantages over the classical Fourier basis in the time-frequency analysis. The need of processing large amount of high dimensional data motivates the extension of the methods based upon the nonlinear Fourier basis to high dimensions. We consider the multi-dimensional nonlinear Fourier basis, which is the tensor product of univariate nonlinear Fourier basis. We investigate the convergence order in norm and also the almost everywhere convergence of the nonlinear Fourier expansions. In order to compute fast and efficiently the nonlinear Fourier expansions of d-dimensional functions, we introduce the sparse nonlinear Fourier expansion and develop a fast algorithm for evaluating it. We also prove that the fast sparse nonlinear Fourier expansions enjoy the optimal convergence order and reduce the computational costs to O ( n log 2 d − 1 n ) . Numerical experiments are presented to demonstrate the efficiency and accuracy of the proposed method.

The fractional Fourier transform (FRFT), a generalization of the Fourier transform, has proven to be a powerful tool in optics and signal processing. Most existing sampling theories of the FRFT consider the class of band-limited signals. However, in the real world, many analog signals encountered in practical engineering applications are non-bandlimited. The purpose of this paper is to propose a sampling theorem for the FRFT, which can provide a suitable and realistic model of sampling and reconstruction for real applications. First, we construct a class of function spaces and derive basic properties of their basis functions. Then, we establish a sampling theorem without band-limiting constraints for the FRFT in the function spaces. The truncation error of sampling is also analyzed. The validity of the theoretical derivations is demonstrated via simulations.

We obtain new inequalities for the Fourier transform in the space L 2 ( R n ) , using a generalized spherical mean operator for proving two estimates in certain classes of functions characterized by a generalized continuity modulus.

We characterize the entire functions of exponential type, whose restriction to the real line is in L p , in different ways: by the usual classical Paley–Wiener growth estimates in the complex plane, by Bernstein inequalities using derivatives or differences, by L p -growth properties of iterated derivatives or differences, and by support properties of the Fourier image. We also establish Paley–Wiener theorems for the Fourier series of a function on the circle.

Fractional Fourier transform (FRFT) plays an important role in many field of optics and signal processing. The recently developed concept of two side fractional quaternion Fourier transform (FRQFT) based on quaternion algebra and FRFT has been found useful for signal and image processing. It is a generalization of the quaternion Fourier transform (QFT). Many properties of the FRQFT are already known, but an extension of the QFT's Plancherel theorem is still missing. The purpose of this paper is to introduce extensions of this theorem. Firstly, we establish scalar value Plancherel theorem for the FRQFT. Then, the concept of right side fractional quaternion Fourier transform (FRQFTr) is proposed. Furthermore, we establish quaternion value Plancherel theorem for the FRQFTr. The Plancherel theorems in QFT domain are shown to be special cases of the achieved results.

In this paper modified variants of the sparse Fourier transform algorithms from Iwen (2010) [32] are presented which improve on the approximation error bounds of the original algorithms. In addition, simple methods for extending the improved sparse Fourier transforms to higher dimensional settings are developed. As a consequence, approximate Fourier transforms are obtained which will identify a near-optimal k-term Fourier series for any given input function, f : [ 0 , 2 π ] D → C , in O ( k 2 ⋅ D 4 ) time (neglecting logarithmic factors). Faster randomized Fourier algorithm variants with runtime complexities that scale linearly in the sparsity parameter k are also presented.

A method for calculating Sobolev smoothed Fourier continuations is presented. The method is based on the recently introduced singular value decomposition based Fourier continuation approach. This approach allows for highly accurate Fourier series approximations of non-periodic functions. These super-algebraically convergent approximations can be highly oscillatory in an extended region, contaminating the Fourier coefficients. It is shown that through solving a subsequent least squares problem, a Fourier continuation can be produced which has been dramatically smoothed in that the Fourier coefficients exhibit a prescribed rate of decay as the wave number increases. While the smoothing procedure has no significant negative effect on the accuracy of the Fourier series approximation, in some situations the smoothed continuations can actually yield increased accuracy in the approximation of the function and its derivatives.

We present an analysis of the convergence of recently developed Fourier continuation techniques that incorporates the required truncation of the Singular Value Decomposition (SVD). Through the analysis, the convergence of SVD-based continuations are related to the convergence of any Fourier approximation of similar form, demonstrating the efficiency and accuracy of the numerical method. The analysis determines that the Fourier continuation approximation error can be bounded by a key value that depends only on the parameters of the Fourier continuation and on the points over which it is applied. For any given distribution of points, a finite number of calculations can be performed to obtain this important value. Our numerical computations on evenly spaced points show that as the number of points increases, this quantity converges to a fixed value, allowing for broad conclusions on the convergence of Fourier continuations calculated with truncated SVDs. We conclude that Fourier continuations can obtain super-algebraic or even exponential convergence on evenly spaced points for non-periodic functions until the convergence is limited by a parameter normally chosen near the machine precision accuracy threshold.

Riemann identities on theta functions are derived using properties of eigenvectors corresponding to the discrete Fourier transform Φ ( 2 ) . In particular we get various fourth order identities of classical Jacobi theta functions.

When multidimensional functions are approximated by a truncated Fourier series, the number of terms typically increases exponentially with the dimension s. However, for functions with more structure than just being L^2-integrable, the contributions from many of the N^s terms in the truncated Fourier series may be insignificant. In this paper we suggest a way to reduce the number of terms by omitting the insignificant ones. We then show how lattice rules can be used for approximating the associated Fourier coefficients, allowing a similar reduction in grid points as in expansion terms. We also show that using a lattice grid permits the efficient computation of the Fourier coefficients by the FFT algorithm. Finally we assemble these ideas into a pseudo-spectral algorithm and demonstrate its efficiency on the Poisson equation.

The central contribution of this paper is the definition of the fractional Fourier transform over finite fields (GFrFT). In order to introduce the GFrFT, concepts related to trigonometry in finite fields are reviewed and some new ideas put forward. In particular, graphic representations of elements in a finite field are suggested and analogies with real and complex numbers are discussed. A modified version of the finite field Fourier transform is given and its eigenstructure is analyzed. This allows us to develop GFrFT theory and investigate its main characteristics. Some illustrative examples are also given throughout the paper.

A generalization of Marcinkiewicz-summability of multi-dimensional Fourier transforms and Fourier series is investigated with the help of a continuous function θ. Under some weak conditions on θ we show that the maximal operator of the Marcinkiewicz-θ-means of a tempered distribution is bounded from H p ( X d ) to L p ( X d ) for all d / ( d + α ) p ⩽ ∞ and, consequently, is of weak type ( 1 , 1 ) , where 0 α ⩽ 1 is depending only on θ and X = R or X = T . As a consequence we obtain a generalization of a summability result due to Marcinkiewicz and Zhizhiashvili for d-dimensional Fourier transforms and Fourier series, more exactly, the Marcinkiewicz-θ-means of a function f ∈ L 1 ( X d ) converge a.e. to f. Moreover, we prove that the Marcinkiewicz-θ-means are uniformly bounded on the spaces H p ( X d ) and so they converge in norm ( d / ( d + α ) p ∞ ) . Similar results are shown for conjugate functions. Some special cases of the Marcinkiewicz-θ-summation are considered, such as the Fejer, Cesaro, Weierstrass, Picar, Bessel, de La Vallee–Poussin, Rogosinski and Riesz summations.

A wide class of physical distributions appears to follow the q-Gaussian form, which plays the role of attractor according to a Central Limit Theorem generalized in the presence of specific correlations between the relevant random variables. In the realm of this theorem, a q-generalized Fourier transform plays an important role. We introduce here a method which univocally determines a distribution from the knowledge of its q-Fourier transform and some supplementary information. This procedure involves a recently q-generalized Dirac delta and the class of functions on which it acts. The present method conveniently extends the inverse of the standard Fourier transform, and is therefore expected to be very useful in the study of many complex systems., 6 pages, 3 figures. To appear in Physics Letters A

New sufficient conditions for the representation of a function via an absolutely convergent Fourier integral are obtained in the paper. In the main result, this is controlled by the behavior near infinity of both the function and its derivative. This result is extended to any dimension d≥2.

The energy method in the Fourier space is useful in deriving the decay estimates for problems in the whole space Rn. In this paper, we study half space problems in R+n=R+×Rn−1 and develop the energy method in the partial Fourier space obtained by taking the Fourier transform with respect to the tangential variable x′∈Rn−1. For the variable x1∈R+ in the normal direction, we use L2 space or weighted L2 space. We apply this energy method to the half space problem for damped wave equations with a nonlinear convection term and prove the asymptotic stability of planar stationary waves by showing a sharp convergence rate for t→∞. The result obtained in this paper is a refinement of the previous one in Ueda et al. (2008) [13].

We prove that the Riemann functional equation can be recovered by the Mellin transforms of essentially all the absolutely integrable functions. The present analysis shows also that the Riemann functional equation is equivalent to the Fourier inversion formula. We introduce the notion of a λ-pair of absolutely integrable functions and show that the components of the λ-pair satisfy an identity involving convolution type products.

Firstly, we study the uniform convergence of cosine and sine Fourier transforms. Secondly, we obtain Pitt–Boas type results on L p -integrability of Fourier transforms with the power weights. The solutions of both problems are written as criteria in terms of general monotone functions.

We use the method of induction-on-scales to prove certain diffeomorphism-invariant nonlinear Brascamp–Lieb inequalities. We provide applications to multilinear convolution inequalities and the restriction theory for the Fourier transform, extending to higher dimensions recent work of Bejenaru–Herr–Tataru and Bennett–Carbery–Wright.

Numerical methods using the Hartley transform are described for the simulation of 3-D wave phenomena with application to the modeling of seismic data. Four topics are covered. The first deals with the solution of the 3-D acoustic wave equation. The second handles the solution of the 3-D two way nonreflecting wave equation. The third involves modeling with an areal source. The fourth treats wave phenomena whose direction of propagation is restricted within ± 90° from a given axis. The numerical methods developed here are similar to the Fourier methods. Time stepping is performed with a second-order differencing operator. The difference is that expressions including space derivative terms are computed by the Hartley transforms rather than the Fourier transforms. Being a real-valued function and equivalent to the Fourier transform, the Hartley transform avoids computational redundancies in terms of the number of operations and memory requirements and thus is more efficient and economical than the Fourier transform. These features are crucial when dealing with 3-D seismic data. The numerical results agree with the analytical results. The use of areal source in modeling can efficiently provide data for testing some schemes that deal with the areal shot-records. Using the transform methods, we can impose constraints on the direction of the wave propagation most precisely in the wavenumber domain when attempting to restrict propagation to upward moving waves. The implementation of the methods is demonstrated on numerical examples.

The aim of this work is to generalize the methods of Fourier Analysis in order to apply them to a wide class of possibly non-integrable functions, with infinitely many variables. The method consists in distinguishing several levels of observability, with a natural meaning. Mathematical coherence is ensured by the fact that these natural concepts are represented within a sure mathematical framework, that of the relative set theory [Y. Peraire, Theorie relative des ensembles internes, Osaka J. Math. 29 (1992) 267–297; Y. Peraire, Some extensions of the principles of idealization transfer and choice in the relative internal set theory, Arch. Math. Logic 34 (1995) 269–277]. This work is also a step for another approach of the Fourier transform of functionals. It can be related to the one, which use double extensions of standard real numbers, performed by T. Nitta and T.Okada in [T. Okada, T. Nitta, Infinitesimal Fourier transformation for the space of functionals, in: Topics in Almost Hermitian Geometry and Related Fields, World Sci. Publ., 2005; T. Okada, T. Nitta, Poisson summation formula for the space of functionals, Nihonkai Math. J. 16 (2005) 1–21].

We consider the problem of reconstructing a compactly supported function with singularities either from values of its Fourier transform available only in a bounded interval or from a limited number of its Fourier coefficients. Our results are based on several observations and algorithms in [G. Beylkin, L. Monzón, On approximation of functions by exponential sums, Appl. Comput. Harmon. Anal. 19 (1) (2005) 17–48]. We avoid both the Gibbs phenomenon and the use of windows or filtering by constructing approximations to the available Fourier data via a short sum of decaying exponentials. Using these exponentials, we extrapolate the Fourier data to the whole real line and, on taking the inverse Fourier transform, obtain an efficient rational representation in the spatial domain. An important feature of this rational representation is that the positions of its poles indicate location of singularities of the function. We consider these representations in the absence of noise and discuss the impact of adding white noise to the Fourier data. We also compare our results with those obtained by other techniques. As an example of application, we consider our approach in the context of the kernel polynomial method for estimating density of states (eigenvalues) of Hermitian operators. We briefly consider the related problem of approximation by rational functions and provide numerical examples using our approach.

For more than a decade it has been empirically known that the causal portion of the inverse Fourier transform of the magnitude spectrum of the speech signal behaves like a minimum-phase signal. Later on, this statement has been shown for an all-pole model. In this paper, we consider related results for both discrete-time Fourier and discrete Fourier transforms of arbitrary sequences. We indicate how the presence of aliasing in circular autocorrelation might be detected. The energy concentration property of zero-phase sequences is discussed.

The theorem of sampling formulae has been deduced for band-limited or time-limited signals in the fractional Fourier domain by different authors. Even though the properties and applications of these formulae have been studied extensively in the literature, none of the research papers throw light on the Poisson sum formula and non-band-limited signals associated with the fractional Fourier transform (FrFT). This paper investigates the generalized pattern of Poisson sum formula from the FrFT point of view and derived several novel sum formulae associated with the FrFT. Firstly, the generalized Poisson sum formula is obtained based on the relationship of the FrFT and the Fourier transform; then some new results associated with this novel sum formula have been derived; the potential applications of these new results in estimating the bandwidth and the fractional spectrum shape of a signal in the fractional Fourier domain are also proposed. In addition, the results can be seen as the generalization of the classical results in the Fourier domain.

Fractional Fourier domains form a continuum of domains making arbitrary angles with the time or frequency domains on the time–frequency plane. Signal representations in these domains are related to the fractional Fourier transform (FrFT). In this paper, a new proof on the shift-invariance of linear time–frequency distributions on fractional Fourier domains is given. We show that short-time Fourier transform (STFT) is the unique linear distribution satisfying magnitude-wise shift-invariance in the fractional Fourier domains. The magnitude-wise shift-invariance property in arbitrary fractional Fourier domains distinguishes STFT among all linear time–frequency distributions and simplifies the interpretation of the resultant distribution as shown by numerical examples.

In the present study, a generalized Fourier transform for time harmonic elastic wave propagation in a half space is developed. The generalized Fourier transform is obtained from the spectral representation of the operator derived from the elastic wave equation. By means of the generalized Fourier transform, a volume integral equation method for the analysis of scattered elastic waves is presented. The proposed method is based on the Krylov subspace iteration technique. During the iterative process, the discrete generalized Fourier transform is used, where the derivation of a huge and dense matrix from the volume integral equation is not necessary.

We obtain new inequalities for the Fourier transform, both on Euclidean space, and on non-compact, rank one symmetric spaces. In both cases these are expressed as a gauge on the size of the transform in terms of a suitable integral modulus of continuity of the function. In all settings, the results present a natural corollary: a quantitative form of the Riemann–Lebesgue lemma. A prototype is given in one-dimensional Fourier analysis.

With convolutions, we determine the Fourier transform of u n [ sinh ( u ) ] n when n is a positive integer. Studying the expansion and taking the Fourier transform of [ u [ sinh ( u ) ] ] d [ u tanh ( u ) ] n when n and d are strictly positive integers, we obtain some polynomials and new probability densities related to them.

On etudie des inegalites LP -> L q a poids pour des transformees de Fourier, en particulier on formule une conjecture de Boas traduisant une integrabilite pour des fonctions dans le cas ou le poids est une puissance lorsque l'une des fonctions est monotone et p = q. Nous donnons des versions unidimensionnelles et multidimensionnelles (dans le cas de fonctions radiales) pour p ≥ q ou p ≤ q et pour une classe definie de fonctions generalement monotones.

The concept of fractional quaternion Fourier transform (FRQFT) is defined in this paper, and the reversibility property, linear property, odd-even invariant property, additivity property and other properties are presented. Meanwhile, the fractional quaternion convolution (FRQCV), fractional quaternion correlation (FRQCR) and product theorem are deduced, and their physical interpretations are given as classical convolution, correlation and product theorem. Moreover, the fast algorithms of FRQFT (FFRQFT) are yielded as well. In addition, we have discovered the relationship between the convolution and correlation in the FRQFT domain, so that the convolution and correlation can be implemented via product theorem in the Fourier transform domain using fast Fourier transform (FFT). Our paper proved that the computation complexities of FRQFT, FRQCV and FRQCR are similar to FFT.

We consider the maximal rank-deficient submatrices of Fourier matrices with order a power of a prime number. We do this by considering a hierarchical subdivision of these matrices into low rank blocks. We also explore some connections with the fast Fourier transform (FFT), and with an uncertainty principle for Fourier transforms over finite Abelian groups.

The Fourier coefficients of a smooth K-invariant function on a compact symmetric space M = U / K are given by integration of the function against the spherical functions. For functions with support in a neighborhood of the origin, we describe the size of the support by means of the exponential type of a holomorphic extension of the Fourier coefficients.