Best two‐parameter rational approximation algorithm of given order: a variation of adaptive Fourier decomposition

The concept of mono‐component is widely used in nonstationary signal processing and time‐frequency analysis. In this paper, we construct several classes of complete rational function systems in the Hardy space, whose boundary values are mono‐components. Then, we propose a best approximation algorithm (BAA) based on optimal selections of two parameters in the orthonormal bases according to the approximation error. Effectiveness of BAA is evaluated by comparison experiments with the classical Fourier decomposition algorithm. It is also shown that BAA has promising results for filtering out noises and dealing with real‐world signals. Copyright © 2014 John Wiley & Sons, Ltd.     

Perturbation of the CR fractional Yamabe problem

In this note, we first prove the non‐degeneracy property of extremals for the optimal Hardy–Littlewood–Sobolev inequality on the Heisenberg group, as an application, a perturbation result for the CR fractional Yamabe problem is obtained, this generalizes a classical result of Malchiodi and Uguzzoni 30 .     

A numerical study of the semi‐classical limit for three‐coupled long wave–short wave interaction equations

We study numerically the semi‐classical limit for three‐coupled long wave–short wave interaction equations. The Fourier–Galerkin semi‐discretization is proved to be spectrally convergent in an appropriate energy space. We propose a split‐step Fourier method in the semi‐classical regime with the discussion of the meshing strategy, which is necessary to obtain correct numerical solution. Plane wave solution with weak and strong initial phases, solitary wave solution and Gaussian solution are considered to investigate the semi‐classical limit.     

Special relativistic Fourier transformation and convolutions

In this paper, we use the steerable special relativistic (space‐time) Fourier transform (SFT) and relate the classical convolution of the algebra for space‐time Cl(3,1)‐valued signals over the space‐time vector space , with the (equally steerable) Mustard convolution. A Mustard convolution can be expressed in the spectral domain as the point wise product of the SFTs of the factor functions. In full generality do we express the classical convolution of space‐time signals in terms of finite linear combinations of Mustard convolutions and vice versa the Mustard convolution of space‐time signals in terms of finite linear combinations of classical convolutions.     

Linearized quantum and relativistic Fokker–Planck–Landau equations

After a recent work on spectral properties and dispersion relations of the linearized classical Fokker–Planck–Landau operator [8], we establish in this paper analogous results for two more realistic collision operators: The first one is the Fokker–Planck–Landau collision operator obtained by relativistic calculations of binary interactions, and the second is a collision operator (of Fokker–Planck–Landau type) derived from the Boltzmann operator in which quantum effects have been taken into account. We apply Sobolev–Poincaré inequalities to establish the spectral gap of the linearized operators. Furthermore, the present study permits the precise knowledge of the behaviour of these linear Fokker–Planck–Landau operators including the transport part. Relations between the eigenvalues of these operators and the Fourier‐space variable in a neighbourhood of 0 are then investigated. This study is a first natural step when one looks for solutions near equilibrium and their hydrodynamic limit for the full non‐linear problem in all space in the spirit of several works [3, 6, 20, 2] on the non‐linear Boltzmann equation. Copyright © 2000 John Wiley & Sons, Ltd.     

On Stein's extension operator preserving Sobolev–Morrey spaces

We prove that Stein's extension operator preserves Sobolev–Morrey spaces, that is spaces of functions with weak derivatives in Morrey spaces. The analysis concerns classical and generalized Morrey spaces on bounded and unbounded domains with Lipschitz boundaries in the n‐dimensional Euclidean space.     

The sharp Strichartz and Sobolev–Strichartz inequalities for the fourth‐order Schrödinger equation

In this paper, we will obtain that there exists a maximizer for the non‐endpoint Strichartz inequalities for the fourth‐order Schrödinger equation with initial data in the L²( R ^d) space in all dimensions, and then we obtain a maximizer also for the non‐endpoint Sobolev–Strichartz inequality for the fourth‐order Schrödinger equation with initial data in the homogeneous Sobolev space. Our analysis derived from the linear profile decomposition. Copyright © 2014 John Wiley & Sons, Ltd.     

Uncertainty principle for measurable sets and signal recovery in quaternion domains

The classical uncertainty principle of harmonic analysis states that a nontrivial function and its Fourier transform cannot both be sharply localized. It plays an important role in signal processing and physics. This paper generalizes the uncertainty principle for measurable sets from complex domain to hypercomplex domain using quaternion algebras, associated with the quaternion Fourier transform. The performance is then evaluated in signal recovery problems where there is an interplay of missing and time‐limiting data. Copyright © 2017 John Wiley & Sons, Ltd.     

Stability of non‐constant steady‐state solutions for non‐isentropic Euler–Maxwell system with a temperature damping term

This work is concerned with the periodic problem for compressible non‐isentropic Euler–Maxwell systems with a temperature damping term arising in plasmas. For this problem, we prove the global in time existence of a smooth solution around a given non‐constant steady state with the help of an induction argument on the order of the mixed time‐space derivatives of solutions in energy estimates. Moreover, we also show the convergence of the solution to this steady state as the time goes to the infinity. This phenomenon on the charge transport shows the essential relation of the systems with the non‐isentropic Euler–Maxwell and the isentropic Euler–Maxwell systems. Copyright © 2015 John Wiley & Sons, Ltd.     

On the existence of incompressible current–vortex sheets: study of a linearized free boundary value problem

We study the initial boundary value problem resulting from the linearization of the equations of ideal incompressible magnetohydrodynamics and the jump conditions on the hypersurface of tangential discontinuity (current–vortex sheet) about an unsteady piecewise smooth solution. Under some assumptions on the unperturbed flow, we prove an energy a priori estimate for the linearized problem. Since the so‐called loss of derivatives in the normal direction to the boundary takes place even for the constant coefficients linearized problem, for the variable coefficients problem and non‐planar current–vortex sheets the natural functional setting is provided by the anisotropic weighted Sobolev space W₂^1,σ. The result of this paper is a necessary step to prove the local in time existence of solutions of the original non‐linear free boundary value problem. The uniqueness of the regular solution of this problem follows already from the a priori estimate we obtain for the linearized problem. Copyright © 2005 John Wiley & Sons, Ltd.     

Radially symmetric solutions of the p‐Laplacian with critical Sobolev–Hardy exponent in perforated‐like domain

In this paper, we establish the existence of radially symmetric solutions of the p‐Laplacian with a concave term and critical Sobolev–Hardy exponent in a perforated‐like domain. Copyright © 2011 John Wiley & Sons, Ltd.     

A discrete Fourier transform based on Simpson's rule

Fourier analysis plays a vital role in the analysis of continuous‐time signals. In many cases, we are forced to approximate the Fourier coefficients based on a sampling of the time signal. Hence, the need for a discrete transformation into the frequency domain giving rise to the classical discrete Fourier transform. In this paper, we present a transformation that arises naturally if one approximates the Fourier coefficients of a continuous‐time signal numerically using the Simpson quadrature rule. This results in a decomposition of the discrete signal into two sequences of equal length. We show that the periodic discrete time signal can be reconstructed completely from its discrete spectrum using an inverse transform. We also present many properties satisfied by this transform. Copyright © 2012 John Wiley & Sons, Ltd.     

A new symmetry approach to solve space–time‐dependent KdV systems

In this paper, a new method to solve space–time‐dependent non‐linear equations is proposed. After considering the variable coefficient of a non‐linear equation as a new dependent variable, some special types of space–time‐dependent equations can be solved from corresponding space–time‐independent equations by using the general classical Lie approach. The rich soliton solutions of space–time‐dependent KdV equation and mKdV equation are given with the help of the approach. Copyright © 2004 John Wiley & Sons, Ltd.     

Phaseless inverse discrete Hilbert transform and determination of signals in shift‐invariant space

In this paper, we first address the uniqueness of phaseless inverse discrete Hilbert transform (phaseless IDHT for short) from the magnitude of discrete Hilbert transform (DHT for short). The measurement vectors of phaseless IDHT do not satisfy the complement property, a traditional requirement for ensuring the uniqueness of phase retrieval. Consequently, the uniqueness problem of phaseless IDHT is essentially different from that of the traditional phase retrieval. For the phaseless IDHT related to compactly supported functions, conditions are given in our first main theorem to ensure the uniqueness. A condition on the step size of DHT is crucial for the insurance. The second main theorem concerns the uniqueness related to noncompactly supported functions. It is not the trivial generalization of the first main theorem. Our third main result is on the determination of signals in shift‐invariant spaces by phaseless IDHT. Note that the measurements used for the determination are the DHT magnitudes. They are the approximations to the Hilbert transform magnitudes. Recall that for the existing methods of phaseless sampling (a special phase‐retrieval problem), to determine a signal depends on the exact measurements but not the approximative ones. Therefore, our determination method is essentially different from the traditional phaseless sampling. Numerical simulations are conducted to examine the efficiency of phaseless IDHT and its application in determining signals in spline Hilbert shift‐invariant space.     

Sobolev‐Jawerth embedding of Triebel‐Lizorkin‐Morrey‐Lorentz spaces and fractional integral operator on Hardy type spaces

A Sobolev type embedding for Triebel‐Lizorkin‐Morrey‐Lorentz spaces is established in this paper. As an application of this result, the boundedness of the fractional integral operator on some generalizations of Hardy spaces such as Hardy‐Morrey spaces and Hardy‐Lorentz spaces are obtained.     

A type theorem on the weighted Herz‐type Hardy spaces and applications

This paper gives a type theorem, which is a boundedness criterion for singular integral operators from the weighted Herz‐type Hardy spaces into the weighted local Herz‐type Hardy spaces. As applications, the corresponding mapping properties for the Cauchy integral and Calderón's commutators are obtained. In addition, a counter example is shown that neither is a Calderón–Zygmund singular integral operator bounded on the homogeneous local Herz‐type Hardy space, nor bounded on the classical local Hardy space.     

Singular integral operator,Hardy–Morrey space estimates for multilinear operators and Navier–Stokes equations

After establishing the molecule characterization of the Hardy–Morrey space, we prove the boundedness of the singular integral operator and the Riesz potential. We also obtain the Hardy–Morrey space estimates for multilinear operators satisfying certain vanishing moments. As an application, we study the existence and the uniqueness of the solutions to the Navier–Stokes equations for the initial data in the Hardy–Morrey space ????(p?n) for q as small as possible. Here, the Hardy–Morrey space estimates for multilinear operators are important tools. Copyright © 2010 John Wiley & Sons, Ltd.     

Uncertainty principles associated with the offset linear canonical transform

As a time‐shifted and frequency‐modulated version of the linear canonical transform (LCT), the offset linear canonical transform (OLCT) provides a more general framework of most existing linear integral transforms in signal processing and optics. To study simultaneous localization of a signal and its OLCT, the classical Heisenberg's uncertainty principle has been recently generalized for the OLCT. In this paper, we complement it by presenting another two uncertainty principles, ie, Donoho‐Stark's uncertainty principle and Amrein‐Berthier‐Benedicks's uncertainty principle, for the OLCT. Moreover, we generalize the short‐time LCT to the short‐time OLCT. We likewise present Lieb's uncertainty principle for the short‐time OLCT and give a lower bound for its essential support.     

Function theory in real Hardy spaces

We show that many classical results in Hardy space theory have exact analogues when the Fourier coefficients are restricted to be real. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

The locally uniformly non‐square points of Orlicz–Bochner sequence spaces

In this paper, we investigate the locally uniformly non‐square point of Orlicz–Bochner sequence spaces endowed with Luxemburg norm. Analysing and combining the generating function M and properties of the real Banach space X , we get sufficient and necessary conditions of locally uniformly non‐square point, which contributes to criteria for locally uniform non‐squareness in Orlicz–Bochner sequence spaces. The results generalize the corresponding results in the classical Orlicz sequence spaces.