On the Hartley – Fourier sine generalized convolution

In this paper, we construct and study a new generalized convolution (f * g)(x) of functions f,g for the Hartley (H₁,H₂) and the Fourier sine (F_s) integral transforms. We will show that these generalized convolutions satisfy the following factorization equalities: We prove the existence of this generalized convolution on different function spaces, such as . As examples, applications to solve a type of integral equations and a type of systems of integral equations are presented. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

基于Hartley变换的三维真振幅偏移算子研究

在实际应用中，以快速Fourier变换为基础的偏移方法，将本来是实数的地震道转化为复数参加运算，导致了计算机内存的增加。本文把只有纯实数运算的Hartley变换引入到基于Fourier变换的偏移算法，再利用三维真振幅偏移单程波方程，结合Fourier变换与Hartley变换的内在关系，经过数学推理，具体导出了裂步Hartley变换真振幅偏移算子。与一般裂步Fourier法相比，裂步Hartley变换真振幅偏移算法既提高了计算效率又对球面扩散问题进行了振幅补偿。     

On the Sobolev boundedness results of the product of pseudo-differential operators involving a couple of fractional Hankel transforms

This article is concerned with the study of pseudo-differential operators associated with fractional Hankel transform. The product of two fractional pseudo-differential operators is defined and investigated its basic properties on some function space. It is shown that the pseudo-differential operators and their products are bounded in Sobolev type spaces. Particular cases are discussed.     

On Numerical Solution of the Markov Renewal Equation: Tight Upper and Lower Kernel Bounds

We develop tight bounds and a fast parallel algorithm to compute the Markov renewal kernel. Knowledge of the kernel allows us to solve Markov renewal equations numerically to study non-steady state behavior in a finite state Markov renewal process. Computational error and numerical stability for computing the bounds in parallel are discussed using well-known results from numerical analysis. We use our algorithm and computed bounds to study the expected number of departures as a function of time for a two node overflow queueing network.     

On the boundedness result of wavelet transform associated with fractional Hankel transform

This article is concerned with the study of the continuity of wavelet transform involving fractional Hankel transform on certain function spaces. The n-dimensional boundedness property of the fractional wavelet transform is also discussed on Sobolev type space. Particular cases are also considered.     

A Widom like formula for some Toeplitz plus Hankel determinants

A well known Widom formula expresses the determinant of a Toeplitz matrix _Tn$T_n$ with Laurant polynomial symbol f in terms of the zeros of f. We give similar formulae for some even Toeplitz plus Hankel matrices. The formulae are based on an analytic representation of the determinant of such matrices in terms of Chebyshev polynomials.     

New convolutions and their applicability to integral equations of Wiener-Hopf plus Hankel type

We propose four new convolutions exhibiting convenient factorization properties associated with two finite interval integral transformations of Fourier-type together with their norm inequalities. Moreover, we study the solvability of a class of integral equations of Wiener-Hopf plus Hankel type (on finite intervals) with the help of the factorization identities of such convolutions. Fourier-type series are used to produce the solution formula of such equations, and a Shannon-type sampling formula is also obtained.     

A discrete convolution involving Fourier sine and cosine series and its applications

ABSTRACT

In this paper, we introduce a discrete convolution involving both the Fourier sine and cosine series. We study Young's type inequality and a discrete transform related to this convolution and solve in closed form a class of discrete Toeplitz plus Hankel equations.     

On multiple integrals of special form

Multiple integrals generalizing the iterated kernels of integral operators are expressed as single integrals in the case of a special representation of the kernel (this is our theorem). Besides integral equations, Markov processes involve these integrals as well. As a consequence of the theorem, we obtain transition probability densities of certain Markov processes. As an illustration, we consider nine examples.     

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.     

On the Toeplitz pencil conjecture

The validity of a conjecture on a singular Toeplitz pencil over ?[x] stated by Schmale and Sharma in connection with an old problem in control theory is presented in this article. The aim of this work is to give a proof of this conjecture.     

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.      

On a class of jointly hyponormal Toeplitz operators

We characterize when a pair of Toeplitz operators is jointly hyponormal under various assumptions--for example, is analytic or is a trigonometric polynomial or is analytic. A typical characterization states that is jointly hyponormal if and only if an algebraic relation of and holds and the single Toeplitz operator is hyponormal, where is a combination of and . More general results for an -tuple of Toeplitz operators are also obtained.

     

On Besov spaces in the Hankel setting

Summary We present new properties of the Besov--Hankel spaces introduced in [10]. We prove a Hankel version of a result of Bui, Paluszyński and Taibleson obtaining new norms that define the topology of the Besov--Hankel spaces. Also we get atomic representations for the distributions in the spaces under consideration.     

On Ptak's Generalization of Hankel Operators

In 1997 Ptak defined generalized Hankel operators as follows: Given two contractions and , an operator is said to be a generalized Hankel operator if and X satisfies a boundedness condition that depends on the unitary parts of the minimal isometric dilations of T ₁ and T ₂. This approach, call it (P), contrasts with a previous one developed by Ptak and Vrbova in 1988, call it (PV), based on the existence of a previously defined generalized Toeplitz operator. There seemed to be a strong but somewhat hidden connection between the theories (P) and (PV) and we clarify that connection by proving that (P) is more general than (PV), even strictly more general for some T ₁ and T ₂, and by studying when they coincide. Then we characterize the existence of Hankel operators, Hankel symbols and analytic Hankel symbols, solving in this way some open problems proposed by Ptak.     

On Hankel Transform and Hankel Convolution of Beurling Type Distributions Having Upper Bounded Support

In this paper we study Beurling type distributions in the Hankel setting. We consider the space of Beurling type distributions on (0, ) having upper bounded support. The Hankel transform and the Hankel convolution are studied on the space . We also establish Paley Wiener type theorems for Hankel transformations of distributions in .     

Bergman空间上可交换的Hankel算子和Toeplitz算子

证明了Bergman空间上的两个小Hankel算子如果是可交换的且其中一个是拟齐次的小Hankel算子,则另一个也是拟齐次的.还研究了Toeplitz算子和小Hankel算子的交换性.