Hankel Kernel Research Articles

Totally positive kernels, Pólya frequency functions, and their transforms

The composition operators preserving total non-negativity and total positivity for various classes of kernels are classified, following three themes. Letting a function act by post composition on kernels with arbitrary domains, it is shown that such a composition operator maps the set of totally non-negative kernels to itself if and only if the function is constant or linear, or just linear if it preserves total positivity. Symmetric kernels are also discussed, with a similar outcome. These classification results are a byproduct of two matrix-completion results and the second theme: an extension of A. M. Whitney’s density theorem from finite domains to subsets of the real line. This extension is derived via a discrete convolution with modulated Gaussian kernels. The third theme consists of analyzing, with tools from harmonic analysis, the preservers of several families of totally non-negative and totally positive kernels with additional structure: continuous Hankel kernels on an interval, Pólya frequency functions, and Pólya frequency sequences. The rigid structure of post-composition transforms of totally positive kernels acting on infinite sets is obtained by combining several specialized situations settled in our present and earlier works.

Recursive moment computation in Filon methods and application to high-frequency wave scattering in two dimensions

Abstract We study the efficient approximation of highly oscillatory integrals using Filon methods. A crucial step in the implementation of these methods is the accurate and fast computation of the Filon quadrature moments. In this work we demonstrate how recurrences can be constructed for a wide class of oscillatory kernel functions, based on the observation that many physically relevant kernel functions are in the null space of a linear differential operator whose action on the Filon interpolation basis is represented by a banded (infinite) matrix. We discuss in further detail the application to two classes of particular interest: integrals with algebraic singularities and stationary points and integrals involving a Hankel function. We provide rigorous stability results for the moment computation for the first of these classes and demonstrate how the corresponding Filon method results in an accurate approximation at truly frequency-independent cost. For the Hankel kernel, we derive error estimates that describe the convergence behaviour of the method in terms of frequency and number of Filon quadrature points. Finally, we show how Filon methods with recursive moment computation can be applied to efficiently compute integrals arising in hybrid numerical-asymptotic collocation methods for high-frequency wave scattering on a screen.

Efficient algorithms for integrals with highly oscillatory Hankel kernels

We present an efficient numerical algorithm for approximating integrals with highly oscillatory Hankel kernels. First, using the integral expression of the Hankel function, the integral is transformed into the integral of the trigonometric kernel, and the singularities are separated. According to whether the integration interval contains zeros or not, the integrals are divided into two cases: singular highly oscillatory integrals and non-singular highly oscillatory integrals. For the non-singular case, the Clenshaw–Curtis–Filon rules (CCF) method is used for fast calculation. For the second case, by using the Cauchy integral theorem, the integral is transformed into an infinite integral, which can be calculated by generalized Gaussian–Laguerre rules. The efficiency and accuracy of the new method are verified by numerical examples.

Moment Estimates of the Cloud of a Planar Measure

With a proper function theoretic definition of the cloud of a positive measure with compact support in the real plane, a computational scheme of transforming the moments of the original measure into the moments of the uniformly distributed mass on the cloud is described. The main limiting operation involves exclusively truncated Christoffel-Darboux kernels, while error bounds depend on the spectral asymptotics of a Hankel kernel belonging to the Hilbert-Schmidt class.

Numerical Steepest Descent Method for Hankel Type of Hypersingular Oscillatory Integrals in Electromagnetic Scattering Problems

We present a fast and accurate numerical scheme for approximating hypersingular integrals with highly oscillatory Hankel kernels. The main idea is to first change the integration path by Cauchy’s theorem, transform the original integral into an integral on a , + ∞ , and then use the generalized Gauss Laguerre integral formula to calculate the corresponding integral. This method has the advantages of high-efficiency, fast convergence speed. Numerical examples show the effect of this method.

Hankel transform, Langlands functoriality and functional equation of automorphic L-functions

This is a survey on recent works of Langlands’s work on functoriality conjectures and related works including the works of Braverman and Kazhdan on the functional equation of automorphic L-functions. Efforts have been made to carry out in complete generality the construction of the L-monoid, and certain a kernel which is, we believe, related to the elusive Hankel kernel.

On the Numerical Quadrature of Weakly Singular Oscillatory Integral and its Fast Implementation

In this paper, we present a Clenshaw-Curtis-Filon-type method for the weakly singular oscillatory integral with Fourier and Hankel kernels. By interpolating the non-oscillatory and nonsingular part of the integrand at $(N+1)$ Clenshaw-Curtis points, the method can be implemented in $O(N \log N)$ operations, which requires the accurate computation of modified moments. We first give a method for the derivation of recurrence relation for the modified moments, which can be applied to the derivation of recurrence relation for the modified moments corresponding to other type oscillatory integrals. By using the recurrence relation, special functions and classic quadrature methods, the modified moments can be computed accurately and efficiently. Then, we present the corresponding error bound in inverse powers of frequencies $k$ and $\omega$ for the proposed method. Numerical examples are provided to support the theoretical results and show the efficiency and accuracy of the method.

An Application of the Weighted Mean Value Method to Fredholm integral equations with Toeplitz plus Hankel kernels

In the present work, we study Fredholm integral equations of the second kind with Toeplitz, Hankel, and Toeplitz plus Hankel kernels. The weighted mean-value method is employed and extended to obtain numerical solutions for one-and multidimensional Fredholm integral equations of the second kind. By a successful application of this method, we first convert the integral equation into a system of algebraic equations, then solve this system and obtain promising results. Finally, numerical examples are provided to show the simplicity and applicability of the method.

Efficient computation of highly oscillatory integrals with Hankel kernel

In this paper, we consider the evaluation of two kinds of oscillatory integrals with a Hankel function as kernel. We first rewrite these integrals as the integrals of Fourier-type. By analytic continuation, these Fourier-type integrals can be transformed into the integrals on [0, +∞), the integrands of which are not oscillatory, and decay exponentially fast. Consequently, the transformed integrals can be efficiently computed by using the generalized Gauss–Laguerre quadrature rule. Moreover, the error analysis for the presented methods is given. The efficiency and accuracy of the methods have been demonstrated by both numerical experiments and theoretical results.

A reproducing kernel Hilbert space constructive approximation for a special class of integral equations

A reproducing kernel Hilbert space approach is proposed to study a class of integral equations with Toeplitz and Hankel kernel functions. The existence property and approximate representations of the solutions are given by constructing appropriate auxiliary operators and positive definite matrices within a reproducing kernel Hilbert space framework. Moreover, conditions for the boundedness and uniqueness of the solution are also obtained.

The finite Hartley new convolutions and solvability of the integral equations with Toeplitz plus Hankel kernels

The main aim of this work is to consider integral equations of convolution type with the Toeplitz plus Hankel kernels firstly posed by Tsitsiklis and Levy (1981) [11]. By constructing eight new generalized convolutions for the finite Hartley transforms we obtain a necessary and sufficient condition for the solvability and unique explicit L2-solution of those equations. Thanks to this convolution approach the solvability condition obtained here is remarkably different from those in Tsitsiklis and Levy (1981) [11] and in other papers.

A new and self‐contained presentation of the theory of boundary operators for slit diffraction and their logarithmic approximations

AbstractWe present a new and self‐contained theory for mapping properties of the boundary operators for slit diffraction occurring in Sommerfeld's diffraction theory, covering two different cases of the polarisation of the light. This theory is entirely developed in the context of the boundary operators with a Hankel kernel and not based on the corresponding mixed boundary value problem for the Helmholtz equation. For a logarithmic approximation of the Hankel kernel we also study the corresponding mapping properties and derive explicit solutions together with certain regularity results.

On the polyconvolution for the Fourier cosine, Fourier sine, and Kontorovich–Lebedev integral transforms

The polyconvolution $ \mathop {*}\limits_1 \left( {f,g,h} \right)(x) $ of three functions f, g, and h is constructed for the Fourier cosine (F c ), Fourier sine (F s ), and Kontorovich–Lebedev (K iy ) integral transforms whose factorization equality has the form $$ {F_c}\left( {\mathop {*}\limits_1 \left( {f,g,h} \right)} \right)(y) = \left( {{F_s}f} \right)(y).\left( {{F_s}g} \right)(y).\left( {{K_{iy}}h} \right)\,\,\,\,\forall y > 0. $$ The relationships between this polyconvolution, the Fourier convolution, and the Fourier cosine convolution are established. In addition, we also establish the relationships between the product of the new polyconvolution and the products of the other known types of convolutions. As an application, we consider a class of integral equations with Toeplitz and Hankel kernels whose solutions can be obtained with the help of the new polyconvolution in the closed form. We also present the applications to the solution of systems of integral equations.

The solvability and explicit solutions of two integral equations via generalized convolutions

This paper presents the necessary and sufficient conditions for the solvability of two integral equations of convolution type; the first equation generalizes from integral equations with the Gaussian kernel, and the second one contains the Toeplitz plus Hankel kernels. Furthermore, the paper shows that the normed rings on L 1 ( R d ) are constructed by using the obtained convolutions, and an arbitrary Hermite function and appropriate linear combination of those functions are the weight-function of four generalized convolutions associating F and F ˇ . The open question about Hermitian weight-function of generalized convolution is posed at the end of the paper.

On the Polyconvolution with the Weight Function for the Fourier Cosine, Fourier Sine, and the Kontorovich-Lebedev Integral Transforms

The polyconvolution with the weight functionγof three functionsf,g, andhfor the integral transforms Fourier sine(Fs), Fourier cosine(Fc), and Kontorovich-Lebedev(Kiy), which is denoted by∗γ(f,g,h)(x), has been constructed. This polyconvolution satisfies the following factorization propertyFc(∗γ(f,g,h))(y)=sin y(Fsf)(y)⋅(Fcg)(y)⋅(Kiyh)(y), for ally>0. The relation of this polyconvolution to the Fourier convolution and the Fourier cosine convolution has been obtained. Also, the relations between the polyconvolution product and others convolution product have been established. In application, we consider a class of integral equations with Toeplitz plus Hankel kernel whose solution in closed form can be obtained with the help of the new polyconvolution. An application on solving systems of integral equations is also obtained.

Diffraction of light revisited

AbstractThe diffraction of monochromatic light is considered for a plane screen with an open infinite slit by solving the vectorial Maxwell–Helmholtz system in the upper half‐space with the Fourier method. With this approach we can represent each solution satisfying an appropriate energy condition by its boundary fields in the Sobolev spaces H±1/2. We show that Sommerfeld's theory using a boundary integral equation with Hankel kernels for the so‐called B‐polarization is covered by our approach, but in general it violates a necessary energy condition. Our representation includes also solutions which are not covered by Sommerfeld's theory. Copyright © 2007 John Wiley & Sons, Ltd.

The dual integral equation method in hydromechanical systems

Some hydromechanical systems are investigated by applying the dual integral equation method. In developing this method we suggest from elementary appropriate solutions of Laplace′s equation, in the domain under consideration, the introduction of a potential function which provides useful combinations in cylindrical and spherical coordinates systems. Since the mixed boundary conditions and the form of the potential function are quite general, we obtain integral equations with mth‐order Hankel kernels. We then discuss a kind of approximate practicable solutions. We note also that the method has important applications in situations which arise in the determination of the temperature distribution in steady‐state heat‐conduction problems.

A Characterization of the range of a finite convolution operator with a hankel kernel

We consider the operator M of convolving a function g supported in the finite interval [-1,1] with the Hankel function . We derive a characterization of the range of M, considered as an operator on LP ([—1,1]), in terms of the Hilbert transform. This characterization can be used as a starting point for solving the integral equation Mg=f with f contained in the range of M. The characterization of the range of M is obtained by applying contour integration methods in the complex plane and it can be expected that by applying such techniques also the ranges of other integral operators can be characterized

Toeplitz and Hankel kernels for estimating time-varying spectra of discrete-time random processes

For a nonstationary random process, the dual-time correlation function and the dual frequency Loeve spectrum are complete theoretical descriptions of second-order behavior. That is, each may be used to synthesize the random process itself, according to the Cramer-Loeve spectral representation. When suitably transformed on one of its two variables, each of these descriptions produces a time-varying spectrum. This spectrum is, in fact, the expected value of the Rihaczek distribution. In this paper, we derive two large families of estimators for this spectrum: one based on a diagonal-Toeplitz-diagonal (dTd) factorization of smoothing kernels and the other based on a diagonal-Hankel-diagonal (dHd) factorization. The dTd factorization produces noncoherent averages of the time-varying spectrogram, and the dHd factorization produces coherent averages. Some of the dTd estimators may be called time-varying power spectrum estimators, and some of the dHd estimators may be called time-varying Wigner-Ville (WV) estimators. The former may always be implemented as multiwindow spectrum estimators, and in some casts, they are true time variations on the Blackman-Tukey-Rosenblatt-Grenander (BTGR) spectrogram. The latter are variations on the Stankovic class of WV estimators.

Null space distributions?A new approach to finite convolution equations with a Hankel kernel

In some earlier publications it has been shown that the solutions of the boundary integral equations for some mixed boundary value problems for the Helmholtz equation permit integral representations in terms of solutions of associated complicated singular algebraic ordinary differential equations. The solutions of these differential equations, however, are required to be known on some infinite interval on the real line, which is unsatisfactory from a practical point of view. In this paper, for the example of one specific boundary integral equation, the relevant solutions of the associated differential equation are expressed by integrals which contain only one unknown generalized function, the support of this generalized function is no longer unbounded but a compact subset of the real line. This generalized function is a distributional solution of the homogeneous boundary integral equation. By this null space distribution the boundary integral equation can be solved for arbitrary right-hand sides, this solution method can be considered of being analogous to the method of variation of parameters in the theory of ordinary differential equations. The nature of the singularities of the null space distribution is worked out and it is shown that the null space distribution itself can be expressed by solutions of the associated ordinary differential equation.