A convolution for the Hankel–Kontorovich–Lebedev transformation

The analysis of a kernel involving the product of three Bessel functions motivates the introduction of the translation operator and the convolution associated to the Hankel–Kontorovich–Lebedev tranformation, first in a classical framework, and then in certain spaces of generalized functions. The main properties of this convolution are investigated, the more important operational rules are obtained and some applications are shown. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Kontorovich‐Lebedev transform and its associated pseudodifferential operator

In this paper, we obtained some useful estimates for convolution corresponding to Kontorovich‐Lebedev transform (KL‐transform) in Lebesgue space. Some continuity theorems for translation, convolution, and KL‐transform in test function space are discussed. Then an integral representation of pseudodifferential operator involving KL‐transform is found out, and its estimates in Lebesgue space is obtained. At the end, some applications of KL‐transform and its convolution are discussed.     

On the orthogonality of the MacDonald's functions

A proof of an orthogonality relation for the MacDonald's functions with identical arguments but unequal complex lower indices is presented. The orthogonality is derived first via a heuristic approach based on the Mehler–Fock integral transform of the MacDonald's functions, and then proved rigorously using a polynomial approximation procedure.     

Clifford–Fourier transform on hyperbolic space

In this paper, we introduce a new generalization of the Helgason–Fourier transform using the angular Dirac operator on both the hyperboloid and unit ball models. The explicit integral kernels of even dimension are derived. Furthermore, we establish the formal generating function of the even dimensional kernels. In the computations, fractional integration plays a key unifying role. Copyright © 2016 John Wiley & Sons, Ltd.     

On the Rudin–Shapiro transform

The Rudin–Shapiro transform (RST) is a linear transform derived from the remarkable Rudin–Shapiro polynomials discovered in 1951. The transform has the notable property of forming a spread spectrum basis for , i.e. the basis vectors are sequences with a nearly flat power spectrum. It is also orthogonal and Hadamard, and it can be made symmetric. This presentation is partly a tutorial on the RST, partly some new results on the symmetric RST that makes the transform interesting from an applicational point-of-view. In particular, it is shown how to make a very simple O(NlogN) implementation, which is quite similar to the Haar wavelet packet transform.     

Signal moments for the short‐time Fourier transform associated with Hardy–Sobolev derivatives

The short‐time Fourier transform has been shown to be a powerful tool for non‐stationary signals and time‐varying systems. This paper investigates the signal moments in the Hardy–Sobolev space that do not usually have classical derivatives. That is, signal moments become valid for non‐smooth signals if we replace the classical derivatives by the Hardy–Sobolev derivatives. Our work is based on the extension of Cohen's contributions to the local and global behaviors of the signal. The relationship of the moments and spreads of the signal in the time, frequency and short‐time Fourier domain are established in the Hardy–Sobolev space. Copyright © 2014 John Wiley & Sons, Ltd.     

An inversion formula for the Segal–Bargmann transform on a symmetric space of non-compact type

We prove analogs of the heat kernel transform inversion formulae of Golse, Leichtnam and the author [E. Leichtnam, F. Golse, M. Stenzel, Intrinsic microlocal analysis and inversion formulae for the heat equation on compact real-analytic Riemannian manifolds, Ann. Sci. École Norm. Sup. (4) 29 (6) (1996) 669–736. MR1422988 (97h:58153), Theorems 0.3, 0.4] in the setting of a Riemannian symmetric space of Helgason's non-compact type.     

The Bergman–Sce transform for slice monogenic functions

In a recent paper, we showed that the classical Bergman theory admits two possible formulations for the class of slice regular functions with quaternionic values. In the so called formulation of the first kind, we provide a Bergman kernel which is defined on and is a reproducing kernel. In the so called formulation of the second kind, we use the Representation Formula for slice regular functions to define a second Bergman kernel; this time the kernel is still defined on U, but the integral representation of f is based on an integral computed only on and the integral does not depend on , (here denotes the sphere unit of purely imaginary quaternions, and represents the complex plane with imaginary unit I). In this paper, we extend the second formulation of the Bergman theory to the case of slice monogenic functions and we focus our attention on the so‐called Bergman–Sce transform. This integral transform is defined by using the Bergman kernel and the Sce mapping theorem and associates to every slice monogenic function f, an axially monogenic function . Copyright © 2011 John Wiley & Sons, Ltd.     

Generalized Hermitean Clifford–Hermite polynomials and the associated wavelet transform

Clifford analysis is a higher‐dimensional function theory offering a refinement of classical harmonic analysis, which has proven to be an appropriate framework for developing higher‐dimensional continuous wavelet transforms, the construction of the wavelets being based on generalizations to a higher dimension of classical orthogonal polynomials on the real line. More recently, Hermitean Clifford analysis has emerged as a new branch of Clifford analysis, offering yet a refinement of the standard Euclidean case; it focusses on so‐called Hermitean monogenic functions, i.e. simultaneous null solutions of two Hermitean Dirac operators. In this Hermitean setting, Clifford–Hermite polynomials and their associated families of wavelet kernels have been constructed starting from a Rodrigues formula involving both Hermitean Dirac operators mentioned. Unfortunately, the property of the so‐called vanishing moments of the corresponding mother wavelets, ensuring that polynomial behaviour in the analyzed signal is filtered out, is only partially satisfied and has to be interpreted with care, the underlying mathematical reason being the fact that the Hermitean Clifford–Hermite polynomials show a too restrictive structure. In this paper, we will remediate this drawback by considering generalized Hermitean Clifford–Hermite polynomials, involving in their definition homogeneous Hermitean monogenic polynomials. The ultimate goal being the construction of new continuous wavelet transforms by means of these polynomials, we first deeply investigate their properties, amongst which are their connection with the traditional Laguerre polynomials, their structure and recurrence relations. Copyright © 2008 John Wiley & Sons, Ltd.     

On compactness of the velocity field in the incompressible limit of the full Navier–Stokes–Fourier system on large domains

The incompressible limit for the full Navier–Stokes–Fourier system is studied on a family of domains containing balls of the radius growing with a speed that dominates the inverse of the Mach number. It is shown that the velocity field converges strongly to its limit locally in space, in particular, the effect of the sound waves is eliminated by means of the local decay estimates for the acoustic wave equation. Copyright © 2008 John Wiley & Sons, Ltd.     

Some relationships for the double modified generalized analytic function space Fourier–Feynman transform and its applications

In this paper we first introduce the concept of a double modified analytic function space Fourier–Feynman transform using the double modified analytic function space integral. We then proceed to establish the existence of the modified analytic function space Fourier–Feynman transform for all functionals in the Banach algebra. Finally we use this double modified analytic function space transform to explain various physical phenomenon.     

Some remarks on “On the windowed Fourier transform and wavelet transform of almost periodic functions,” by J.R. Partington and B. Ünalmı

J.R. Partington and B. Ünalmı consider in their 2001 paper [J.R. Partington, B. Ünalmı , Appl. Comput. Harmon. Anal. 10 (1) (2001) 45–60] the windowed Fourier transform and wavelet transform as tools for analyzing almost periodic signals. They establish Parseval-type identities and consider discretized versions of these transforms in order to construct generalized frame decompositions. We have found a gap in the construction of generalized frames in the windowed Fourier transform case; we comment on this gap and give an alternative proof. As for the wavelet transform case, in [J.R. Partington, B. Ünalmı , Appl. Comput. Harmon. Anal. 10 (1) (2001) 45–60] the generalized frame decomposition is done only for the simplest wavelet, the Haar wavelet; we show how to construct generalized frame decompositions for a wide family of wavelets.     

On the accuracy of the approximation of the complex exponent by the first terms of its Taylor expansion with applications

A modification of the Taylor expansion for the complex exponential function e^ix$e^{ix}$, x∈R$x\in \mathbf{R}$, is proposed yielding precise moment-type estimates of the accuracy of the approximation of a Fourier transform by the first terms of its Taylor expansion. Moreover, a precise upper bound for the third moment of a probability distribution in terms of the absolute third moment is established. Based on these results, new precise bounds for Fourier–Stieltjes transforms of probability distribution functions and for their derivatives are obtained that are uniform in classes of distributions with prescribed first three moments.     

On the recovery of a 2-D function from the modulus of its Fourier transform

A uniqueness theorem is proven for the problem of the recovery of a complex valued compactly supported 2-D function from the modulus of its Fourier transform. An application to the phase problem in optics is discussed.     

On the Brown–Shields conjecture for cyclicity in the Dirichlet space

Let be the Dirichlet space, namely the space of holomorphic functions on the unit disk whose derivative is square-integrable. We establish a new sufficient condition for a function to be cyclic, i.e. for to be dense in . This allows us to prove a special case of the conjecture of Brown and Shields that a function is cyclic in iff it is outer and its zero set (defined appropriately) is of capacity zero.     

On the existence of finite energy weak solutions to the Navier–Stokes equations in irregular domains

This paper studies the existence of weak solutions of the Navier–Stokes system defined on a certain class of domains in ?³ that may contain cusps. The concept of such a domain and weak energy solution for the system is defined and its existence is proved. However, thinness of cusps must be related to the adiabatic constant appearing in the pressure law. Copyright © 2008 John Wiley & Sons, Ltd.     

Scattering theory for the cubic nonlinear Klein–Gordon system

In this paper, we consider the scattering theory of a nonlinear Klein–Gordon system, which describes the interaction of two scalar fields. The analysis in this paper is an adaptation of the technique used by Nakanishi, which is originally due to Bourgain. The new technical point appears in the localization argument of proving a concentration phenomenon. Copyright © 2013 John Wiley & Sons, Ltd.     

On the convergence of difference schemes for generalized Benjamin–Bona–Mahony equation

We consider an initial boundary‐value problem for the generalized Benjamin–Bona–Mahony equation. A three‐level conservative difference schemes are studied. The obtained algebraic equations are linear with respect to the values of unknown function for each new level. It is proved that the scheme is convergent with the convergence rate of order k – 1, when the exact solution belongs to the Sobolev space of order . © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 301–320, 2014     

On the spectral norms of Cauchy–Toeplitz and Cauchy–Hankel matrices

In this study, we have found upper and lower bounds for the spectral norm of Cauchy–Toeplitz and Cauchy–Hankel matrices in the forms T_n=[1/(a+(i−j)b)]ⁿ_i,j=1, H_n=[1/(a+(i+j)b)]ⁿ_i,j=1.