On real Paley-Wiener theorems for certain integral transforms

We prove real Paley-Wiener theorems for the (inverse) Jacobi transform, characterising the space of L²-functions whose image under the Jacobi transform are (smooth) functions with compact support.     

Hypoelliptic Jacobi–Dunkl Convolution of Distributions

In this paper we give a necessary and sufficient condition in terms of the Jacobi–Dunkl transform in order that a Jacobi–Dunkl convolution of distributions is hypoelliptic.     

Generalized Convolution for the Discrete Jacobi Transformation

We study the translation and the convolution associated to the discrete Jacobi transformation on complex sequences of slow and rapid growth. Also we establish new topological properties for these spaces of sequences. This revised version was published online in June 2006 with corrections to the Cover Date.     

A local two radii theorem on the Chébli-Trimèche hypergroup

Let 0<r₁<r₂<r₁+r₂<R. It is given a necessary and sufficient condition so that the null function is the unique solution f∈C^∞(]−R,+R[) of the system

(1)     

Chébli-Trimèche hypergroups and W-type spaces

The generalized Fourier transformation associated with Chébli-Trimèche hypergroups is investigated in some spaces of W-type introduced by Gelfand and Shilov. It is established that this transformation is an isomorphism from the space W_M,a onto the space W^M×,1/a, where the function M and the parameter a determine the growth of the testing functions in the first space, and M^× denotes the Young dual function of M. The translation operator and the convolution corresponding to this transform are also studied in this class of spaces.     

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.     

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 .     

The Inversion of Multiscale Convolution Approximation and Average of Distributions

We develop two kinds of inversion formulas of the multiscale convolution approximation which is defined by a convolution kernel . The inversion formulas are constructed by a convolution kernel which is defined in terms of and has a vanishing moment of order one. A large class of generalized moving average approximations with B-splines, Box-splines and exponential Box-splines (EB-splines) as convolution kernels is included in the theory formulated in this paper. The average of distributions is considered, and correspondingly, the formulas related to the EB-splines are obtained from the -average.     

A Comparison on the Commutative Neutrix Convolution of Distributions and the Exchange Formula

Let , be ultradistributions in and let and where is a sequence in which converges to the Dirac-delta function . Then the neutrix product is defined on the space of ultradistributions as the neutrix limit of the sequence provided the limit exist in the sense that

Convolution Operators on Discrete Hardy Spaces

We establish the connection between the boundedness of convolution operators on H^p(ℝ^N) and some related operators on H^p(ℤ^N). The results we obtain here extend the already known for L^p spaces with p > 1. We also study similar results for maximal operators given by convolution with the dilation of a fixed kernel. Our main tools are some known results about functions of exponential type already presented in [BC1] that, in particular, allow us to prove a sampling theorem for functions of exponential type belonging to Hardy spaces     

Convolution structure associated with the Jacobi-Dunkl operator on IR

In this paper, a product formula for the eigenfunction of the Jacobi-Dunkl differential-difference operator is derived. It leads to a uniformly bounded convolution of point measures and a signed hypergroup on IR. 2000 Mathematics Subject Classification Primary—34K99, 44A15, 44A35, 43A15     

Stochastic flows acting on Schwartz distributions

We define the compositionT° _s,t of a Schwartz distributionT with a stochastic flow _s,t generated by a stochastic differential equation. Then we establish a generalized Itô's formula for the composite processesT(t)° _s,t andT(t)° _s,t ^–1 , which describe a differential rule with respect to timet. The formula is then applied to two problems. One is the regularity of semigroups induced by the stochastic flow. The other is the existence and the continuity of the local time with respect to the spatial parameter, of a one dimensional stochastic flow.     

广义维纳泛函,白噪声分布及实Schwartz广义函数空间上的分布

在实Ｓｃｈｗａｒｔｚ广义函数空间上,证明了复值广义维纳泛函,由Ｋｏｎｄｒａｔｅｖ－Ｓｔｒｅｉｔ及Ｈｉｄａ构造的复值白噪声分布都是由Ｋｈｒｅｎｎｉｋｏｖ构造的分布。利用上述结果进而证明了,一类无穷维伪微分算子是由复值广义维纳泛函空间上的连续线性算子族扩张而成。更进一步,还证明了由Ｋｈｒｅｎｎｉｋｏｖ构造的关于分布的试验函数空间是关于白噪声泛函的Ｍｅｙｅｒ－Ｙａｎ试验函数空间的子空间。     

Convolution semigroups and central limit theorem associated with a dual convolution structure

We consider hypergroups associated with Jacobi functions ⁽⁾ (x), (–1/2). We prove the existence of a dual convolution structure on [0,+[i(]0,s ₀]{{) =++1,s ₀=min(,–+1). Next we establish a Lévy-Khintchine type formula which permits to characterize the semigroup and the infinitely divisible probabilities associated with this dual convolution, finally we prove a central limit theorem.     

加权Herz型Hardy空间上的强奇异积分算子

本文证明了当α＝ｎ１－１ｑ时，强奇异积分算子是从齐次加权Ｈｅｒｚ型Ｈａｒｄｙ空间ＨＫα，ｐｑ（ｗ１；ｗ２）到齐次加权Ｈｅｒｚ空间Ｋα，ｐｑ（ｗ１；ｗ２）上的有界算子．而且，该算子在非齐次加权Ｈｅｒｚ型Ｈａｒｄｙ空间ＨＫα，ｐｑ（ｗ１；ｗ２）上的有界性也被考查．     

Convolution Operators on Expanding Polyhedra: Limits of the Norms of Inverse Operators and Pseudospectra

We consider matrix convolution operators with integrable kernels on expanding polyhedra. We study their connection with convolution operators on the cones at the vertices of polyhedra. We prove that the norm of the inverse operator on a polyhedron tends to the maximum of the norms of the inverse operators on the cones, and the pseudospectrum tends to the union of the corresponding pseudospectra. The study bases on the local method adapted to this kind of problems.     

强奇异卷积算子交换子的Hardy型空间估计

本文给出了强奇异卷积算子交换子[6,T]在Hardy型空间中的估计,其中 b∈BMO(Rn)且一致连续, T为强奇异卷积算子.     

An integral operational matrix based on Jacobi polynomials for solving fractional-order differential equations

This paper aims to construct a general formulation for the Jacobi operational matrix of fractional integral operator. Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. Therefore, a reliable and efficient technique for the solution of them is too important. For the concept of fractional derivative we will adopt Caputo’s definition by using Riemann–Liouville fractional integral operator. Our main aim is to generalize the Jacobi integral operational matrix to the fractional calculus. These matrices together with the Tau method are then utilized to reduce the solution of this problem to the solution of a system of algebraic equations. The method is applied to solve linear and nonlinear fractional differential equations. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.     

Regularized Solutions of a Nonlinear Convolution Equation on the Euclidean Group

In this paper we apply Fourier analysis on the two and three dimensional Euclidean motion groups to the solution of a nonlinear convolution equation. First, we review the theory of the irreducible unitary representations of the motion group and discuss the corresponding Fourier transform of functions on the motion group. The main reasons why exact solutions of this convolution equation do not exist in many cases are discussed. Techniques for regularization of the problem and numerical methods for finding approximate solutions are presented. Examples are considered and approximate solutions are found.