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We propose a refinement of the Robertson–Schrödinger uncertainty principle (RSUP) using Wigner distributions. This new principle is stronger than the RSUP. In particular, and unlike the RSUP, which can be saturated by many phase space functions, the refined RSUP can be saturated by pure Gaussian Wigner functions only. Moreover, the new principle is technically as simple as the standard RSUP. In addition, it makes a direct connection with modern harmonic analysis, since it involves the Wigner transform and its symplectic Fourier transform, which is the radar ambiguity function. As a by-product of the refined RSUP, we derive inequalities involving the entropy and the covariance matrix of Wigner distributions. These inequalities refine the Shanon and the Hirschman inequalities for the Wigner distribution of a mixed quantum state \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document}ρ. We prove sharp estimates which critically depend on the purity of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document}ρ and which are saturated in the Gaussian case.

In this article, we present the abstract harmonic analysis aspects of the operator-valued continuous Gabor transform (CGT) on second countable, non-unimodular, and type I locally compact groups. We show that the operator-valued continuous Gabor transform CGT satisfies a Plancherel formula and an inversion formula. As an example, we study these results on the continuous affine group.

We study nonuniform sampling in shift-invariant spaces and the construction of Gabor frames with respect to the class of totally positive functions whose Fourier transform factors as $ \hat g(\xi)= \prod_{j=1}^n (1+2\pi i\delta_j\xi)^{-1} \, e^{-c \xi^2}$ for $\delta_1,\ldots,\delta_n\in \mathbb{R}, c >0$ (in which case $g$ is called totally positive of Gaussian type). In analogy to Beurling's sampling theorem for the Paley-Wiener space of entire functions, we prove that every separated set with lower Beurling density $>1$ is a sampling set for the shift-invariant space generated by such a $g$. In view of the known necessary density conditions, this result is optimal and validates the heuristic reasonings in the engineering literature. Using a subtle connection between sampling in shift-invariant spaces and the theory of Gabor frames, we show that the set of phase-space shifts of $g$ with respect to a rectangular lattice $\alpha \mathbb{Z} \times \beta \mathbb{Z}$ forms a frame, if and only if $\alpha \beta, Comment: 25 pages

The deformation of an infinite bar subjected to a self-equilibrated load distribution is investigated using the peridynamic formulation of elasticity theory. The peridynamic theory differs from the classical theory and other nonlocal theories in that it does not involve spatial derivatives of the displacement field. The bar problem is formulated as a linear Fredholm integral equation and solved using Fourier transform methods. The solution is shown to exhibit, in general, features that are not found in the classical result. Among these are decaying oscillations in the displacement field and progressively weakening discontinuities that propagate outside of the loading region. These features, when present, are guaranteed to decay provided that the wave speeds are real. This leads to a one-dimensional version of St. Venant's principle for peridynamic materials that ensures the increasing smoothness of the displacement field remotely from the loading region. The peridynamic result converges to the classical result in the limit of short-range forces. An example gives the solution to the concentrated load problem, and hence provides the Green's function for general loading problems.

In this paper, we define a horospherical transform for a semisimple symmetric space Y. A natural double fibration is used to assign a more geometrical space Ξ of horospheres to Y. The horospherical transform relates certain integrable analytic functions on Y to analytic functions on Ξ by fiber integration. We determine the kernel of the horospherical transform and establish that the transform is injective on functions belonging to the most continuous spectrum of Y.

We give a geometric characterization of interpolating varieties for the algebra of Fourier–Laplace transforms of ultradistributions of Roumieu type with compact support on the real line in the non-quasianalytic case. In particular, such a characterization is found in the case of classical Gevrey classes. We also provide a relation between interpolating varieties in this case and in the case of Hormander algebras related to ultradistributions of Beurling type.

The offset linear canonical transform (OLCT) is the name of a parameterized continuum of transforms which include, as particular cases, the most widely used linear transforms in engineering such as the Fourier transform (FT), fractional Fourier transform (FRFT), Fresnel transform (FRST), frequency modulation, time shifting, time scaling, chirping and others. Therefore the OLCT provides a unified framework for studying the behavior of many practical transforms and system responses. In this paper the sampling theorem for OLCT is considered. The sampling theorem for OLCT signals presented here serves as a unification and generalization of previously developed sampling theorems.

We show that, given a tempered distribution S whose Fourier transform is a function of polynomial growth, a point x in \({\mathbb{R}^{n}}\) is outside the support of S if and only if the Fourier integral of S is summable in Bochner-Riesz means to zero uniformly on a neighbourhood of x.

In the present analysis we introduced a calculation strategy of the staggered grid differentiation by using the real value FFT and real inverted FFT for the pseudospectral method and applied the technique to seismic wave simulation. The calculation method introduced here is one third faster on average than the traditional differentiation method by using the complex FFT. The introduced differentiation strategy is very efficient in economy. For example we apply the staggered grid differentiation by real valued FFT to the simulation of seismic wave propagation in inhomogeneous medium. The results show the validity of the present method.

Nonaxisymmetric solutions to time-fractional diffusion-wave equation with a source term in cylindrical coordinates are obtained for an infinite medium. The solutions are found using the Laplace transform with respect to time , the Hankel transform with respect to the radial coordinate , the finite Fourier transform with respect to the angular coordinate , and the exponential Fourier transform with respect to the spatial coordinate . Numerical results are illustrated graphically.

The aim of this work is to establish in great detail The q-Fourier analysis related to the q-cosine. The wise reader will note that the considered q-cosine coincides with the one given by T.H. Koornwinder and S.F. Swarttouw. Through the q-cosine product formula, we define and analyze the properties of the q-even translation and the q-convolution. Adopting the Titchmarsh approach, we study the q-cosine Fourier transform and its inverse formula.

Two-dimensional elastodynamic displacements and stresses for a monoclinic solid have been obtained in relatively simple form by the use of the eigenvalue method, following Laplace and Fourier transforms. The main aim of this paper is to present a straightforward analytical eigenvalue method for a monoclinic solid which avoids the cumbersome nature of the problem and is convenient for numerical computation. The use of matrix notation avoids unwieldy mathematical expressions. A particular case of normal line-load acting in an orthotropic solid is discussed in detail. The corresponding deformation in time-domain is obtained numerically. The variations of elastodynamic displacements and stresses for an anisotropic medium with the horizontal distance have been shown graphically. It has been found that anisotropy is affecting the trend of distribution curves significantly.

The paper introduces a new objective method for speech quality assessment called Fourier-lapped multilayer perceptron (FLMLP). This method uses an overcomplete transform based on the discrete Fourier transform (DFT) and modulated lapped transform (MLT). This transform generates the DFT and the MLT speech spectral domains from which several relevant perceptual parameters are extracted. The proposed method also employs a multilayer perceptron neural network trained by a modified version of the scaled conjugated gradient method. This neural network maps the perceptual parameters into a subjective score. The numerical results show that FLMLP is an effective alternative to previous methods. As a result, it is worth stating that the techniques here described may be potentially useful to other researches facing the same kind of problem.

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License., A novel Fourier-based technique for local motion detection from image sequences is proposed. In this method, the instantaneous velocities of local image points are inferred directly from the global 3D Fourier components of the image sequence. This is done by selecting those velocities for which the superposition of the corresponding Fourier gratings leads to constructive interference at the image point. Hence, image velocities can be assigned locally even though position is computed from the phases and amplitudes of global Fourier components (spanning the whole image sequence) that have been filtered based on the motion-constraint equation, reducing certain aperture effects typically arising from windowing in other methods. Regularization is introduced for sequences having smooth flow fields. Aperture effects and their effect on optic-flow regularization are investigated in this context. The algorithm is tested on both synthetic and real image sequences and the results are compared to those of other local methods. Finally, we show that other motion features, i.e. motion direction, can be computed using the same algorithmic framework without requiring an intermediate representation of local velocity, which is an important characteristic of the proposed method. © The Author(s) 2010. This article is published with open access at Springerlink.com., The work has received support by the German Ministry for Education and Research (BMBF) via the Bernstein Center for Computational Neuroscience (BCCN) Göttingen under Grant No. 01GQ0430 and the EU project Drivsco under contract number 016276- 2. B.D. also acknowledges support from the Spanish Ministry for Science and Innovation via a Ramon y Cajal Fellowship.

In this paper we develop the theory of Fourier multiplier operators $T_{m}:L^{p}(\mathbb{R}^{d};X)\to L^{q}(\mathbb{R}^{d};Y)$, for Banach spaces $X$ and $Y$, $1\leq p\leq q\leq \infty$ and $m:\mathbb{R}^d\to \mathcal{L}(X,Y)$ an operator-valued symbol. The case $p=q$ has been studied extensively since the 1980's, but far less is known for $p, Comment: Revised version, to appear in Journal of Fourier Analysis and Applications. 31 pages. The results on Besov spaces and the proof of the extrapolation result have been moved to arXiv:1606.03272

We study linear q-difference-differential equations under the action of a perturbation parameter . This work deals with a q-analog of the research made in (Lastra and Malek in Adv. Differ. Equ. 2015:200, 2015) giving rise to a generalization of the work (Malek in Funkc. Ekvacioj, 2015, to appear). This generalization is related to the nature of the forcing term which suggests the use of a q-analog of an acceleration procedure. The proof leans on a q-analog of the so-called Ramis-Sibuya theorem which entails two distinct q-Gevrey orders. The work concludes with an application of the main result when the forcing term solves a related problem.

We treat the phase retrieval (PR) problem of reconstructing the interest signal from its Fourier magnitude. Since the Fourier phase information is lost, the problem is ill-posed. Several techniques have been used to address this problem by utilizing various priors such as non-negative, support, and Fourier magnitude constraints. Recent methods exploiting sparsity are developed to improve the reconstruction quality. However, the previous algorithms of utilizing sparsity prior suffer from either the low reconstruction quality at low oversampled factors or being sensitive to noise. To address these issues, we propose a framework that exploits sparsity of the signal in the translation invariant Haar pyramid (TIHP) tight frame. Based on this sparsity prior, we formulate the sparse representation regularization term and incorporate it into the PR optimization problem. We propose the alternating iterative algorithm for solving the corresponding non-convex problem by dividing the problem into several subproblems. We give the optimal solution to each subproblem, and experimental simulations under the noise-free and noisy scenario indicate that our proposed algorithm can obtain a better reconstruction quality compared to the conventional alternative projection methods, even outperform the recent sparsity-based algorithms in terms of reconstruction quality.

In this paper, we use the Approximation Formula for the Fourier transform of the solution set of lattice points on k-spheres and methods of Bourgain and Ionescu to refine the \(\ell ^p(\mathbb {Z}^d)\)-boundedness results for discrete k-spherical maximal functions to a restricted weak-type result at the endpoint. We introduce a density-parameter, which may be viewed as a discrete version of Minkowski dimension used in related works on the continuous analgoue, in order to exploit recent progress of Wooley and Bourgain–Demeter–Guth on the Vinogradov mean value conjectures via a novel Approximation Formula for a single average, and obtain improved bounds for lacunary discrete k-spherical maximal functions when \(k \ge 3\).

By developing the classical kernel method, Delaigle and Meister provide a nice estimation for a density function with some Fourier-oscillating noises over a Sobolev ball and over risk (Delaigle and Meister in Stat. Sin. 21:1065-1092, 2011). The current paper extends their theorem to Besov ball and risk with by using wavelet methods. We firstly show a linear wavelet estimation for densities in over risk, motivated by the work of Delaigle and Meister. Our result reduces to their theorem, when . Because the linear wavelet estimator is not adaptive, a nonlinear wavelet estimator is then provided. It turns out that the convergence rate is better than the linear one for . In addition, our conclusions contain estimations for density derivatives as well.

In this paper, we mainly study the boundedness of unimodular Fourier multipliers with a time parameter on the modulation spaces where is a differentiable real-valued function, namely we estimate under the multiplier norm, denoted by . The sharpness of s and the regularity lost are also discussed when the multiplier acts on functions in modulation spaces. Meanwhile the lower bound of the multiplier is shown. Finally, we present a discussion of the relationship between the main result and well-posedness results for nonlinear PDEs already existing in the literature. MSC:42B15, 42B35, 42C15.

In this paper, we identify the unknown source which depends only on spatial variable for a fractional diffusion equation using the Fourier method. Not alike the previous literature, we propose to choose the regularization parameter by an a posteriori rule, with which we can obtain error estimate of Hölder type between the exact solution and the regularized approximation. Numerical simulations show that the proposed scheme is effective and stable. MSC: 35R30, 47A52, 65M30, 65M32.

In this paper, we prove the local well-posedness for the incompressible porous media equation in Triebel-Lizorkin spaces and obtain blow-up criterion of smooth solutions. The main tools we use are the Fourier localization technique and Bony’s paraproduct decomposition.

In this paper, using the Young inequality as regards the Herz space, the authors obtain the boundedness of the Fourier transform on the Herz type amalgams and Besov spaces. As corollaries, the authors get the estimates of the Fourier transform on the weighted amalgams and Besov spaces.

The Robin problem for the Helmholtz equation in normal-polar annuli is addressed by using a suitable Fourier-Hankel series technique. Attention is in particular focused on the wide class of domains whose boundaries are defined by the so-called superformula introduced by Gielis. A dedicated numerical procedure based on the computer algebra system Mathematica© is developed in order to validate the proposed methodology. In this way, highly accurate approximations of the solution, featuring properties similar to the classical ones, are obtained.

In this paper, we consider the Fourier spectral method for numerically solving the modified Swift-Hohenberg equation. The semi-discrete and fully discrete schemes are established. Moreover, the existence, uniqueness and the optimal error bound are also considered.

We consider the problem of identifying the pollution source of a 1D parabolic equation from the initial and the final data. The problem is ill posed and regularization is in order. Using the quasi-boundary method and the truncation Fourier method, we present two regularization methods. Error estimates are given and the methods are illustrated by numerical experiments.

We consider the general solution of quartic functional equations and prove the Hyers-Ulam-Rassias stability. Moreover, using the pullbacks and the heat kernels we reformulate and prove the stability results of quartic functional equations in the spaces of tempered distributions and Fourier hyperfunctions.

The Hilbert transform along curves is defined by the principal value integral. The pointwise existence of the principal value Hilbert transform can be educed from the appropriate estimates for the corresponding maximal Hilbert transform. By using the estimates of Fourier transforms and Littlewood-Paley theory, we obtain -boundedness for the maximal Hilbert transform associated to curves , where , P is a real polynomial and γ is convex on . Then, we can conclude that the Hilbert transform along curves exists in pointwise sense. MSC:42B20, 42B25.