A criterion for a two-sided integral transform to be unitary

A criterion for a two-sided Watson transform to be unitary in the space L₂(R) is considered. It enables us to construct new examples of integral transforms with symmetric inversion formulas (only the Fourier and the Hartley transforms are known). We give some new examples of the indicated transforms, in particular, the symmetric Hankel transform with the sum of two Bessel functions in the kernal and the Hardy transform with the sum of a Neumann function and a Struve function in the kernel, and the Narain transform with a sum of two G-functions in the kernelDoctor of Physicomathematical sciences.Candidate of Physicomathematical sciences.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 5, pp. 697–699, May, 1992.     

An integral transform whose kernel is a stream function of a steady Euler flow with axisymmetry

A new integral transform whose kernel is a Stokes stream function of an exact solution to the axisymmetric stationary Euler equations is derived with its inverse by using a Coulomb wave function. In the process of its derivation, a set of integral transforms with respect to the radial variable is also found. Each of them is different from the Hankel (the Fourier–Bessel) transform.     

An integral transform whose kernel is a stream function of a steady Euler flow with axisymmetry

A new integral transform whose kernel is a Stokes stream function of an exact solution to the axisymmetric stationary Euler equations is derived with its inverse by using a Coulomb wave function. In the process of its derivation, a set of integral transforms with respect to the radial variable is also found. Each of them is different from the Hankel (the Fourier–Bessel) transform.     

Continuous and discrete Bessel wavelet transforms

Using the theory of Hankel convolution, continuous and discrete Bessel wavelet transforms are defined. Certain boundedness results and inversion formula for the continuous Bessel wavelet transform are obtained. Important properties of the discrete Bessel wavelet transform are given.     

A new construction of the Clifford-Fourier kernel

In this paper, we develop a new method based on the Laplace transform to study the Clifford-Fourier transform. First, the kernel of the Clifford-Fourier transform in the Laplace domain is obtained. When the dimension is even, the inverse Laplace transform may be computed and we obtain the explicit expression for the kernel as a finite sum of Bessel functions. We equally obtain the plane wave decomposition and find new integral representations for the kernel in all dimensions. Finally we define and compute the formal generating function for the even dimensional kernels.     

The Class of Clifford-Fourier Transforms

Recently, there has been an increasing interest in the study of hypercomplex signals and their Fourier transforms. This paper aims to study such integral transforms from general principles, using 4 different yet equivalent definitions of the classical Fourier transform. This is applied to the so-called Clifford-Fourier transform (see Brackx et al., J. Fourier Anal. Appl. 11:669–681, 2005). The integral kernel of this transform is a particular solution of a system of PDEs in a Clifford algebra, but is, contrary to the classical Fourier transform, not the unique solution. Here we determine an entire class of solutions of this system of PDEs, under certain constraints. For each solution, series expressions in terms of Gegenbauer polynomials and Bessel functions are obtained. This allows to compute explicitly the eigenvalues of the associated integral transforms. In the even-dimensional case, this also yields the inverse transform for each of the solutions. Finally, several properties of the entire class of solutions are proven.     

Finite Interval Convolution Operators with Transmission Property

We study convolution operators in Bessel potential spaces and (fractional) Sobolev spaces over a finite interval. The main purpose of the investigation is to find conditions on the convolution kernel or on a Fourier symbol of these operators under which the solutions inherit higher regularity from the data. We provide conditions which ensure the transmission property for the finite interval convolution operators between Bessel potential spaces and Sobolev spaces. These conditions lead to smoothness preserving properties of operators defined in the above-mentioned spaces where the kernel, cokernel and, therefore, indices do not depend on the order of differentiability. In the case of invertibility of the finite interval convolution operator, a representation of its inverse is presented in terms of the canonical factorization of a related Fourier symbol matrix function.     

New index transforms of the Lebedev–Skalskaya type

New index transforms, involving the real part of the modified Bessel function of the first kind as the kernel are considered. Mapping properties such as the boundedness and invertibility are investigated for these operators in the Lebesgue spaces. Inversion theorems are proved. As an interesting application, a solution of the initial value problem for the second-order partial differential equation, involving the Laplacian, is obtained. It is noted that the corresponding operators with the imaginary part of the modified Bessel function of the first kind lead to the familiar Kontorovich–Lebedev transform and its inverse.     

On the Fourier transform of SO(d)-finite measures on the unit sphere

The paper presents a simple new approach to the problem of computing Fourier transforms of SO(d)-finite measures on the unit sphere in the euclidean space. Representing such measures as restrictions of homogeneous polynomials we use the canonical decomposition of homogeneous polynomials together with the plane wave expansion to derive a formula expressing such transforms under two forms, one of which was established previously by F. J. Gonzalez Vieli. We showthat equivalence of these two forms is related to a certain multi-step recurrence relation for Bessel functions, which encompasses several classical identities satisfied by Bessel functions. We show it leads further to a certain periodicity relation for the Hankel transform, related to the Bochner- Coifman periodicity relation for the Fourier transform. The purported novelty of this approach rests on the systematic use of the detailed form of the canonical decomposition of homogeneous polynomials, which replaces the more traditional approach based on integral identities related to the Funk-Hecke theorem. In fact, in the companion paper the present authors were able to deduce this way a fairly general expansion theorem for zonal functions, which includes the plane wave expansion used here as a special case.Received: 7 May 2004; revised: 11 October 2004     

Convolution Hankel Transforms on the Zemanian Spaces

In this paper we investigate the convolution Hankel transforms on the Zemanian spaces of Hankel transformable functions and distributions. The convolution Hankel transform is defined on generalized functions by using the adjoint method. Our new definition includes as special cases other known definitions of the convolution Hankel transform of distributions. Finally we establish a distributional inversion formula for the transformation under consideration involving Bessel differential operators.     

On the index integral transformation with Nicholson's function as the kernel

The integral transformation, which is associated with the Nicholson function as the kernel, is introduced and investigated in the paper. This transformation is an integral, where integration is with respect to an index of the sum of squares of Bessel functions of the first and second kind. Composition representations and relationships with the Meijer K-transform, the Kontorovich-Lebedev transform, the Mellin transform, and the sine Fourier transform are given. We also present boundedness properties, a Parseval type equality, and an inversion formula.     

On the Hankel transformation of order zero

In this paper we write the Hankel transform of order zero of a function as a composition of Fourier transforms, and a new proof is given for Hankel's theorem on Hankel transformation of order zero.     

A discrete Hartley transform based on Simpson's rule

The Hartley transform is an integral transformation that maps a real valued function into a real valued frequency function via the Hartley kernel, thereby avoiding complex arithmetic as opposed to the Fourier transform. Approximation of the Hartley integral by the trapezoidal quadrature results in the discrete Hartley transform, which has proven a contender to the discrete Fourier transform because of its involutory nature. In this paper, a discrete transform is proposed as a real transform with a convolution property and is an alternative to the discrete Hartley transform. Copyright © 2015 John Wiley & Sons, Ltd.     

Method of Local Peak Functions for Reconstructing the Original Profile in the Fourier Transformation

We propose a method for reconstructing the original profile function in the one-dimensional Fourier transformation from the module of the Fourier transform function analytically. The major concept of the method consists in representing the modeling profile function as a sum of local peak functions. The latter are chosen as eigenfunctions generated by linear differential equations with polynomial coefficients. This allows directly inverting the Fourier transformation without numerical integration. The solution of the inverse problem thus reduces to a nonlinear regression with a small number of optimizing parameters and to a numerical or asymptotic study of the corresponding modeling peak functions taken as the eigenfunctions of the differential equations and their Fourier transforms.     

Closed form of the Fourier–Borel Kernel in the Framework of Clifford Analysis

In this paper we derive for the even dimensional case a closed form of the Fourier–Borel kernel in the Clifford analysis setting. This kernel is obtained as the monogenic component in the Fischer decomposition of the exponential function ${e^{\langle \underline{x}, \underline{u} \rangle}}$ where ${\langle . , . \rangle}$ denotes the standard inner product on the m-dimensional Euclidean space. A first approach based on Clifford analysis techniques leads to a conceptual formula containing the Gamma operator and the so-called Clifford–Bessel function, two fundamental objects in the theory of Clifford analysis. To obtain an explicit expression for the Fourier–Borel kernel in terms of a finite sum of Bessel functions, this formula remains however hard to work with. To that end we have also elaborated a more direct approach based on special functions leading to recurrence formulas for a closed form of the Fourier–Borel kernel.     

Continuous vs. discrete fractional Fourier transforms

We compare the finite Fourier (-exponential) and Fourier–Kravchuk transforms; both are discrete, finite versions of the Fourier integral transform. The latter is a canonical transform whose fractionalization is well defined. We examine the harmonic oscillator wavefunctions and their finite counterparts: Mehta's basis functions and the Kravchuk functions. The fractionalized Fourier–Kravchuk transform was proposed in J. Opt. Soc. Amer. A (14 (1997) 1467–1477) and is here subject of numerical analysis. In particular, we follow the harmonic motions of coherent states within a finite, discrete optical model of a shallow multimodal waveguide.     

On a Certain Class of Integral Equations Associated with Hankel Transforms

This paper deals with a new class of Fredholm integral equations of the first kind associated with Hankel transforms of integer order. Analysis of the equations is based on operators transforming Bessel functions of the first kind into kernels of Weber–Orr integral transforms. Their inverse operators are established by means of new inversion theorems for the Hankel and Weber–Orr integral transforms of functions belonging to L ¹ and L ². These operators together with the proven Paley–Wiener’s theorem for the Weber–Orr transform enable to regularize the equations and, in special cases, to derive explicit solutions. The integral equations analyzed in this paper can be employed instead of dual integral equations usually treated with the Cooke–Lebedev method. An example manifests that it may be preferable because of the possibility to control norms of operators in the regularized equations.      

Nonreflecting boundary condition for the wave equation

To solve the time-dependent wave equation in an infinite two (three) dimensional domain a circular (spherical) artificial boundary is introduced to restrict the computational domain. To determine the nonreflecting boundary we solve the exterior Dirichlet problem which involves the inverse Fourier transform. The truncation of the continued fraction representation of the ratio of Hankel function, that appear in the inverse Fourier transform, provides a stable and numerically accurate approximation. Consequently, there is a sequence of boundary conditions in both two and three dimensions that are new. Furthermore, only the first derivatives in space and time appear and the coefficients are updated in a simple way from the previous time step. The accuracy of the boundary conditions is illustrated using a point source and the finite difference solution to a Dirichlet problem.     

若干周期再生核空间的覆盖数

借助离散Fourier变换给出估计Mercer核矩阵逆矩阵范数上界的一种方法,由此给出了估计周期再生核Hilbert空间覆盖数的上、下界的一般方法.特别, 对两种特殊的周期再生核空间覆盖数的上、下界进行了比较.     

A note on the unsteady flow of a generalized second-grade fluid through a circular cylinder subject to a time dependent shear stress

The unsteady flow of a generalized second-grade fluid through an infinite straight circular cylinder is considered. The flow of the fluid is due to the longitudinal time dependent shear stress that is prescribed on the boundary of the cylinder. The fractional calculus approach in the governing equation corresponding to a second-grade fluid is introduced. The velocity field and the resulting shear stress are obtained by means of the finite Hankel and Laplace transforms. In order to avoid lengthy calculations of residues and contour integrals, the discrete inverse Laplace transform method is used. The corresponding solutions for ordinary second-grade and Newtonian fluids, performing the same motion, are obtained as limiting cases of our general solutions. Finally, the influence of the material constants and of the fractional parameter on the velocity and shear stress variations is underlined by graphical illustrations.