Hankel Transform Research Articles

The Hankel transform of the sum of consecutive generalized Catalan numbers

We discuss the properties of the Hankel transformation of a sequence whose elements are the sums of consecutive generalized Catalan numbers and find their values in the closed form.

Poroelastic model for pile–soil interaction in a half‐space porous medium due to seismic waves

AbstractIn this paper, frequency domain dynamic response of a pile embedded in a half‐space porous medium and subjected to P, SV seismic waves is investigated. According to the fictitious pile methodology, the problem is decomposed into an extended poroelastic half‐space and a fictitious pile. The extended porous half‐space is described by Biot's theory, while the fictitious pile is treated as a bar and a beam and described by the conventional 1‐D structure vibration theory. Using the Hankel transformation method, the fundamental solutions for a half‐space porous medium subjected to a vertical or a horizontal circular patch load are established. Based on the obtained fundamental solutions and free wave fields, the second kind of Fredholm integral equations describing the vertical and the horizontal interaction between the pile and the poroelastic half‐space are established. Solution of the integral equations yields the dynamic response of the pile to plane P, SV waves. Numerical results show the parameters of the porous medium, the pile and incident waves have direct influences on the dynamic response of the pile–half‐space system. Significant differences between conventional single‐phase elastic model and the poroelastic model for the surrounding medium of the pile are found. Copyright © 2007 John Wiley & Sons, Ltd.

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.

Dynamic response of a pile embedded in a porous medium subjected to plane SH waves

In this paper, the frequency domain dynamic response of a pile embedded in a porous medium subjected to SH seismic waves is investigated. The surrounding porous medium of the pile is described by Biot’s poro-elastic theory, while the pile embedded in the porous medium is treated as a beam and described by a beam vibration theory. Using the Hankel transformation method, the fundamental solution for a half-space porous medium subjected to a horizontal circular patch load is established. According to the fictitious pile methodology, the second kind of Fredholm integral equation for the pile is established in terms of the obtained fundamental solution and free wave field. The solution of the integral equation yields the dynamic response of the pile to plane SH waves. Numerical results indicate that the parameters of the porous medium, the pile and incident waves have considerable influences on the dynamic response of the pile and the porous medium.

Spacetime Causality in the Study of the Hankel Transform

We study Hilbert space aspects of the Klein-Gordon equation in two-dimensional spacetime. We associate to its restriction to a spacelike wedge a scattering from the past light cone to the future light cone, which is then shown to be (essentially) the Hankel transform of order zero. We apply this to give a novel proof, solely based on the causality of this spatio-temporal wave propagation, of the theorem of de Branges and V. Rovnyak concerning Hankel pairs with a support property. We recover their isometric expansion as an application of Riemann’s general method for solving Cauchy-Goursat problems of hyperbolic type.

Far-field radiated from a vertical magnetic dipole in the sea with a rough upper surface

The problem of communication in the sea has been considered as propagation of radio waves in a three-layered medium (air, sea, and ground). With the aid of the perturbation calculus, this paper analyzes the influence exerted onto the electromagnetic field of arrangements radiating a pure transverse electric field in the sea. The sea height varies continuously with the distance from the transmitting end due to sea waves. Knowledge of the solution for the case of uniform sea height is presumed. The problem of the transition conditions at the upper boundary of the sea is solved in the imaging space of a Hankel transformation. Its reversal produces an integral presentation of the interfering field, which was previously quite difficult to evaluate. In this study, closed-form expressions for the far-field of a vertical magnetic dipole embedded below the sea surface are obtained by using a new technique to evaluate this integral with the aid of the complex image theory. The results obtained are compared with those mentioned elsewhere

Second Order Diffraction of Water Waves by a Bottom Mounted Vertical Circular Cylinder and Some Related Numerical Problems

A Hankel transformation is used to obtain the second order diffraction solution of vertical cylinder of circular cross section. The improper integral over the free surface is tackled carefully. The singularity at the free surface is overcome effectively using a third order nonlinear transformation. Numerical results for free surface elevations compare well with the published data.

Fast computation of Hankel Transform using orthonormal exponential approximation of complex kernel function

The computation of electromagnetic (EM) fields, for 1-D layered earth model, requires evaluation of Hankel Transform (HT) of the EM kernel function. The digital filtering is the most widely used technique to evaluate HT integrals. However, it has some obvious shortcomings. We present an alternative scheme, based on an orthonormal exponential approximation of the kernel function, for evaluating HT integrals. This approximation of the kernel function was chosen because the analytical solution of HT of an exponential function is readily available in literature. This expansion reduces the integral to a simple algebraic sum. The implementation of such a scheme requires that the weights and the exponents of the exponential function be estimated. The exponents were estimated through a guided search algorithm while the weights were obtained using Marquardt matrix inversion method. The algorithm was tested on analytical HT pairs available in literature. The results are compared with those obtained using the digital filtering technique with Anderson filters. The field curves for four types (A-, K-, H-and Q-type) of 3-layer earth models are generated using the present scheme and compared with the corresponding curves obtained using the Anderson sc heme. It is concluded that the present scheme is more accurate than the Anderson scheme

A Generalization of the Hankel Transform and the Lorentz Multipliers

Let $\phi$ be a bounded function on $[0,\infty)$ continuous except on a null set, and $\phi_{\epsilon}(\xi)=\phi(\epsilon\xi)\ (\epsilon>0).$ Also let $\tilde{T}_{\epsilon}$ be the operator on Jacobi series such that $(\tilde{T}_{\epsilon}f)^{\wedge}(n)=\phi_{\epsilon}(n)\hat{f}(n)\ (n\in{\bf Z})$, where $\hat{f}(n)$ is the coefficient of Jacobi expanstion of $f$, and ${\cal H}_{\alpha}(Tf)(\xi)$ be defined by $\phi(\xi){\cal H}_{\alpha}f(\xi)\ (\xi\in(0,\infty))$, where ${\cal H}_{\alpha}f$ is the modified Hankel transform of $f$ with order $\alpha$. Then the author [7] proved that if the operator norm of $\tilde{T}_{\epsilon}$ is uniformly bounded for all $\epsilon>0$, $T$ is a bounded operator on the modified Hankel transforms in the Lorentz spaces, and we have the maximal type theorem in the Lorentz spaces, respectively. In this paper, we give a generalized definition of the modified Hankel transform and the Hankel transform, and prove a generalization of the results in [7].

Study Hankel Transforms and Properties of Bessel Function via Entangled StateRepresentation Transformation in Quantum Mechanics

In Phys. Lett. A 313 (2003) 343 we have found that theself-reciprocal Hankel transformation (HT) is embodied in quantummechanics by a transform between two entangled state representationsof continuum variables. In thiswork we study Hankel transforms and properties of Bessel function viaentangled state representations' transformation in quantum mechanics.

Lp-Norm inequality of the product of pseudo-differential operators involving Hankel transformation

Lp-norm type inequalities for pseudo-differential operators L(x,D) and H(x,D) are derived for 1 ≤ p < ∞. Product of pseudo-differential operators associated with Bessel operator Sμ are defined and its Lp-norm type inequalities are obtained.

CR1Dmod: A Matlab program to model 1D complex resistivity effects in electrical and electromagnetic surveys

We have constructed a forward modelling code in Matlab, capable of handling several commonly used electrical and electromagnetic methods in a 1D environment. We review the implemented electromagnetic field equations for grounded wires, frequency and transient soundings and present new solutions in the case of a non-magnetic first layer. The CR1Dmod code evaluates the Hankel transforms occurring in the field equations using either the Fast Hankel Transform based on digital filter theory, or a numerical integration scheme applied between the zeros of the Bessel function. A graphical user interface allows easy construction of 1D models and control of the parameters. Modelling results are in agreement with other authors, but the time of computation is less efficient than other available codes. Nevertheless, the CR1Dmod routine handles complex resistivities and offers solutions based on the full EM-equations as well as the quasi-static approximation. Thus, modelling of effects based on changes in the magnetic permeability and the permittivity is also possible.

Face recognition based on polar frequency features

A novel biologically motivated face-recognition algorithm based on polar frequency is presented. Polar frequency descriptors are extracted from face images by Fourier--Bessel transform (FBT). Next, the Euclidean distance between all images is computed and each image is now represented by its dissimilarity to the other images. A pseudo-Fisher linear discriminant was built on this dissimilarity space. The performance of discrete Fourier transform (DFT) descriptors and a combination of both feature types was also evaluated. The algorithms were tested on a 40- and 1196-subjects face database (ORL and FERET, respectively). With five images per subject in the training and test datasets, error rate on the ORL database was 3.8, 1.25, and 0.2% for the FBT, DFT, and the combined classifier, respectively, as compared to 2.6% achieved by the best previous algorithm. The most informative polar frequency features were concentrated at low-to-medium angular frequencies coupled to low radial frequencies. On the FERET database, where an affine normalization preprocessing was applied, the FBT algorithm outperformed only the PCA in a rank recognition test. However, it achieved performance comparable to state-of-the-art methods when evaluated by verification tests. These results indicate the high informative value of the polar frequency content of face images in relation to recognition and verification tasks and that the Cartesian frequency content can complement information about the subjects' identity, but possibly only when the images are not prenormalized. Possible implications for human face recognition are discussed.

Integral Equations for the Spin-Weighted Spheroidal WaveFunctions

We present new integral equations for the spin-weighted spheroidal wavefunctions which in turn should lead to global uniform estimates andshould help in particular in the study of their dependence on theparameters. For the prolate spheroidal wavefunction with m = 0, thereexists the integral equation whose kernel is (sin x)/x, andthe sinc function kernel (sin x)/x is of great mathematicalsignificance. We also extend the similar sinc function kernel(sin x)/x to the case m≠0 and s≠0, whichinterestingly turn out as some kind of Hankel transformations.

Mathematical Model for Biological Vesicle in Osmophoretic Migration

Osmophoresis indicates the motion of biological cells or small capsules in concentration gradients. This study presents a mathematical model that combined analytical- numerical study for the osmophoretic migration of a spherical biological vesicle perpendicular to two parallel plates. The imposed concentration gradient is constant and perpendicular to two parallel plates. The biological vesicle, which is a body of fluid enveloped by a continuous semipermeable membrane, may hold arbitrary solute and is assumed to maintain its spherical shape. The existence of the plane walls result in two basic effects on the biological vesicle velocity: first, the walls increase viscous retardation of the moving biological vesicle; secondly, the local concentration gradient on the biological vesicle surface is enhanced by the walls, therefore speeding up the biological vesicle. In order to solve the concentration and hydrodynamic governing equations, general solutions are constructed from the fundamental solutions in both the circular cylindrical and spherical coordinate systems. The boundary conditions are enforced first at the plane wall by the Hankel transforms and then on the biological vesicle surface by means of a boundary collocation method. For special cases of osmophoretic migration of a spherical biological vesicle normal a single plane wall, the collocation result of numerical simulations agree well with the solutions obtained by using spherical bipolar coordinates. In general, the existence of the walls always give rise to an enhancement in the osmophoretic migration.

Spaces of Generalized Type ${\cal H}_{\mu}$, Spaces of Type $S$ and the Hankel Transformation

Spaces of Generalized Type ${\cal H}_{\mu}$, Spaces of Type $S$ and the Hankel Transformation

Formula omitted]-Boundedness of localization operators associated to Bessel left regular representations

Motivated from Wong [M.W. Wong, L p -Boundedness of localization operators associated to left regular representations, Proc. Amer. Math. Soc. 130 (2002) 2911–2929], an L μ p -boundedness of localization operators associated to Bessel's left regular representations of locally compact Hausdorff group is obtained and an application of Bessel wavelet multipliers is investigated by using Zemanian theory of Hankel transformation.

Torsional oscillations of a rigid disc bonded to multilayered poroelastic medium

This paper deals mainly with the dynamic response of a rigid disc bonded to the surface of a layered poroelastic half-space. The disc is subjected to time-harmonic torsional moment loadings. The half space under consideration consists of a number of layers with different thickness and material properties. Hankel transform techniques and transferring matrix method are used to solve the governing equations. The continuity of the displacement and stress fields between different layers enabled derivation of closed-form solutions in the transform domain. On the assumption that the contact between the disc and the half space is perfectly bonded, this dynamic mixed boundary-value problem can be reduced to dual integral equations, which are further reduced to Fredholm integral equations of the second kind and solved by numerical procedures. Selected numerical results for the dynamic impedance and displacement amplitude of the disc resting on different saturated models are presented to show the influence of the material and geometrical properties of both the saturated soil-foundation system and the nature of the load acting on it. The conclusions obtained can serve as guidelines for practical engineering.

Study on the three-dimensional theory of super thin multi-layered films irradiated by the continuously modulated laser

On the basis of heat conduction equations and under special boundary conditions,the temperatute fields of multilayered heated uniaxial films can be acquired in the cylindrical coordinate system by Hankel transformation and retransformation.As an example with specific parameters,the special growth conditions for three-layered films are discussed.The larger are the lognitudinal heat conductions vs transverse ones of thin films for higher thermal diffusivity,the larger are chan ges of phases of the temperature fields in the thin films.However,it is faintly influencing for the lognitudinal heat conductions vs transverse ones of thin fil ms for lower thermal diffusivity,and the smaller is light absorbing coefficient, the more evident is the influence on phases because of frequency change.

The steady motion of a circular crack along the interface of a layer and a half-space

An exact analytical solution is given of the direct axisymmetric steady dynamic problem in the theory of elasticity of the motion, with an arbitrary constant subsonic velocity ν = c along the interface of a rigidly coupled layer 0 ≤ z ≤ H and a half-space z ≤ 0, of a circular transverse shear crack 0 ≤ r ≤ r0 + ct (r0 ≤ 0) with and without a cavity at its tip. Using Hankel transformation in terms of biwave potentials, a general solution of the fundamental equations of motion in the theory of elasticity and the basic solutions of the first fundamental boundary-value problem are separately constructed for the layer and the half-space for the case of arbitrary normal and shear stresses in the plane of separation z = 0 in a moving cylindrical system of coordinates r1 = r + ct, z1 = z. A special regularization of the main solution is carried out which ensures the convergence of the integrals for all stresses and displacements while preserving the high accuracy of the solution to whatever level may be desired [1, 2]. On the basis of the main solutions, a mathematical formulation is given of the mixed problem of the motion of a transverse shear crack with a cavity at the tip and its reduction to a system of three singular integral equations with Cauchy kernels which allows of regularization by the Carleman-Vekua method in terms of the closed solution of the corresponding characteristic system of singular integral equations. When the width of the cavity vanishes, one of the equations of the system solves the problem of a transverse shear crack without a cavity. Criteria are established for the existence of a cavity and its absence as a function of the elastic and velocity characteristics of the layer and half-space and the velocity of motion of the crack c.