On the calculation of the finite Hankel transform eigenfunctions

In this work, we discuss the numerical computation of the eigenvalues and eigenfunctions of the finite (truncated) Hankel transform, important for numerous applications. Due to the very special behavior of the Hankel transform eigenfunctions, their direct numerical calculation often causes an essential loss of accuracy. Here, we present several simple, efficient and robust numerical techniques to compute Hankel transform eigenfunctions via the associated singular self-adjoint Sturm-Liouville operator. The properties of the proposed approaches are compared and illustrated by means of numerical experiments.     

Spectral analysis of the finite Hankel transform and circular prolate spheroidal wave functions

In this paper, we develop two practical methods for the computation of the eigenvalues as well as the eigenfunctions of the finite Hankel transform operator. These different eigenfunctions are called circular prolate spheroidal wave functions (CPSWFs). This work is motivated by the potential applications of the CPSWFs as well as the development of practical methods for computing their values. Also, in this work, we should prove that the CPSWFs form an orthonormal basis of the space of Hankel band-limited functions, an orthogonal basis of L²([0,1]) and an orthonormal system of L²([0,+∞[). Our computation of the CPSWFs and their associated eigenvalues is done by the use of two different methods. The first method is based on a suitable matrix representation of the finite Hankel transform operator. The second method is based on the use of an efficient quadrature method based on a special family of orthogonal polynomials. Also, we give two Maple programs that implement the previous two methods. Finally, we present some numerical results that illustrate the results of this work.     

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.     

Numerical Projection Method For Inverse Fourier Transform And Its Application

Numerical projection method of the Fourier transform inversion from data given on a finite interval is proposed. It is based on an expansion of the solution into a series of eigenfunctions of the Fourier transform. The number of terms of the expansion depends on the length of the data interval. Convergence of the solution of the method is proved. The projection method for the case of the sine Fourier transform and the set of the odd Hermite functions being its eigenfunctions are examined and applied to numerical Fourier filtering.     

Dirichlet空间上Toeplitz算子与Berezin型变换相关的一些性质

本文研究了单位圆盘D 的Dirichlet 空间上Toeplitz 算子和小Hankel 算子. 利用Berezin 型变换讨论了Toeplitz 算子的不变子空间问题, 具有Berezin 型符号的Toeplitz 算子的渐进可乘性以及Toeplitz 算子的Riccati 方程的可解性. 应用Berezin 变换得到了Toeplitz 算子和小Hankel 算子可逆的充分条件. 此外, 还利用Hankel 算子和Berezin 变换刻画了算子2T_uv-T_uT_v-T_vT_u 的紧性, 其中函数u,v ∈ L^2,1.     

Direct form seminorms arising in the theory of interpolation by Hankel translates of a basis function

Certain spaces of functions which arise in the process of interpolation by Hankel translates of a basis function, as developed by the authors elsewhere, are defined with respect to a seminorm which is given in terms of the Hankel transform of each function. This kind of seminorm is called an indirect one. Here we discuss essentially two cases in which the seminorm can be rewritten in direct form, that is, in terms of the function itself rather than its Hankel transform. This is expected to lead to better estimates of the interpolation error.     

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.     

Compact Hankel Forms on Planar Domains

A Hankel form on a Hilbert function space is a bounded, symmetric, bilinear form [., .] satisfying [fx, y] = [x, fy] for a class of multipliers f. We prove analogs of Weyl–Horn and Ky Fan inequalities for compact Hankel forms, and apply them to estimate the related eigenvalues, both for Hardy–Smirnov and Bergman spaces norms associated to multiply connected planar domains. In the case of the unit disk, we investigate the asymptotic of some measures constructed by eigenfunctions of Hankel operators with certain Markov functions as symbols. Submitted: May 2, 2008. Accepted: June 28, 2008.     

Real Paley-Wiener Theorems for the Hankel Transform

We first characterize the image of the compactly supported smooth even functions under the Hankel transform as a subspace of the Schwartz space. We then describe the space of smooth L^p-functions whose Hankel transform has compact support as a subspace of the space of L^p-functions.     

Generalized Integral Transforms with the Homotopy Perturbation Method

This paper applies He’s homotopy perturbation method to compute a large variety of integral transforms. The Esscher, Fourier, Hankel, Laplace, Mellin and Stieljes integrals transforms are particular cases of our generalized integral transform. Our method is of practical importance in order to derive new integration formulae, to approximate certain difficult integrals as well as to calculate the expectation of certain nonlinear functions of random variable.     

Tables of Fourier, Laplace and Hankel Transforms of (u,n)-Dimensional Generalized Functions

The present note contains the Tables of Fourier, Laplace and Hankel transforms of several dimensional generalized functions. They are, mainly, based on the Laplace transform of retarded, Lorentz-invariant functions and the Fourier transforms of causal distributions.     

A Modification of the Projection Method for Integral Equation of the First Kind

A modification of the projection method is proposed for an integral equation of the first kind with a Fourier core on an interval. The proposed method replaces the eigenvectors corresponding to a multiple eigenvalue with odd Hermite functions — the eigenfunctions of the Fourier sine transform on the half-line.     

Distributional Chébli-Trimèche transforms

In this paper we investigate the distributional Chébli-Trimèche transforms. We use the so-called kernel method and we are inspired by the papers of Dube and Pandey [L.S. Dube, J.N. Pandey, On the Hankel transform of distributions, Tôhoku Math. J. 27 (1975) 337-354] and Koh and Zemanian [E.L. Koh, A.H. Zemanian, The complex Hankel and I-transformations of generalized functions, SIAM J. Appl. Math. 16 (1968) 945-957] about distributional Hankel transforms. We note that our procedure, supported in a representation of the elements in the corresponding dual spaces, is simpler than the methods described in the above mentioned papers. Some applications of our distributional theory are presented.     

Biorthogonal decompositions of the Radon transform

Summary. The concept of singular value decompositions is a valuable tool in the examination of ill-posed inverse problems such as the inversion of the Radon transform. A singular value decomposition depends on the determination of suitable orthogonal systems of eigenfunctions of the operators , . In this paper we consider a new approach which generalizes this concept. By application of biorthogonal instead of orthogonal functions we are able to apply a larger class of function sets in order to account for the structure of the eigenfunction spaces. Although it is preferable to use eigenfunctions it is still possible to consider biorthogonal function systems which are not eigenfunctions of the operator. With respect to the Radon transform for functions with support in the unit ball we apply the system of Appell polynomials which is a natural generalization of the univariate system of Gegenbauer (ultraspherical) polynomials to the multivariate case. The corresponding biorthogonal decompositions show some advantages in comparison with the known singular value decompositions. Vice versa by application of our decompositions we are able to prove new properties of the Appell polynomials. Received October 19, 1993     

On distributional finite hankel transform

A new characterization of the finite hankel transform for the generalized functions is developed, using families of regular generalized functions generated by smooth functions of slow growth. An inversion formula is established, in the distributional sense, using the new characterization. This new characterization is equivalent to Zemanian's extension of the Hankel transform to distributions obtained from the generalization of the Parseval's equation     

Using empirical Eigenfunctions and Galerkin method to two‐phase transport models

In this article, we consider a nonlinear partial differential system describing two‐phase transports and try to recover the source term and the nonlinear diffusion term when the state variable is known at different profile times. To this end, we use a POD‐Galerkin procedure in which the proper orthogonal decomposition technique is applied to the ensemble of solutions to derive empirical eigenfunctions. These empirical eigenfunctions are then used as basis functions within a Galerkin method to transform the partial differential equation into a set of ordinary differential equations. Finally, the validation of the used method has been evaluated by some numerical examples. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 456–474, 2007     

Continuity of the fractional Hankel wavelet transform on the spaces of type S

In this present article, we study the fractional Hankel transform and its inverse on certain Gel'fand‐Shilov spaces of type S. The continuous fractional wavelet transform is defined involving the fractional Hankel transform. The continuity of fractional Hankel wavelet transform is discussed on Gel'fand‐Shilov spaces of type S. This article goes further to discuss the continuity property of fractional Hankel transform and fractional Hankel wavelet transform on the ultradifferentiable function spaces.     

非对称载荷作用的外部圆形裂纹问题

使用边界积分方程方法,研究了三维无限弹性体中受非对称载荷作用的外部圆形裂纹问题。通过使用Fourier级数和超几何函数,将问题的二维边界奇异积分方程简化为Abel型方程,获得了一般非对称载荷作用的外部圆形裂纹问题的应力强度因子精确解,比用Hankel变换法得到的结果更为一般。结果表明:边界积分方程法在解析分析方面还有很大的潜力。     

Unsteady flow of a dusty gas in cylinders with circular and sectorial cross sections

The problems of flow of a viscous incompressible gas embedded with particles in a circular cylinder and a cylinder whose cross section is a sector of a circle are discussed. The analysis applies to flows with pressure gradients which are arbitrary functions of time. Explicit expressions for the exact velocities of the gas and particles are obtained by using the methods of operational calculus—the finite Hankel transform, the Laplace transform, etc. Numerical results are obtained for developing flow due to a constant pressure gradient.     

Recurrent calculations of multipole matrix elements

We propose a new method for calculating multipole matrix elements between wave eigenfunctions of the one-dimensional Schrödinger equation. The method is based on the transition to the auxiliary third- and fourth-order equations, to which an analogue of the Laplace transform is then applied. The resulting recursive procedure allows us to evaluate matrix elements starting with a number of eigenvalues that are assumed to be known and several basis matrix elements. As an example, we consider the multipole matrix elements between the wave functions of the harmonic and nonharmonic oscillators.