The Cross-Correlation Operation of Schwartz Distributions

A generalization of the cross-correlation operation of the Schwartz distributions is considered, and some properties of this operation are established.     

广义函数多目标规划的充分条件

本文借助于广义函数的调和表示，采用非标准分析方法，给出广义函数多目标规划的几个充分条件．     

Hypoelliptic Jacobi Convolution Operators on Schwartz Distributions

In this paper we characterize the hypoellipticity of Jacobi convolution operators on Schwartz distributions. In the proof of the main result of this paper the positivity ofthe convolution structure for the inverse of the Jacobi transform plays an essential role. We also study hypoelliptic convolution equations on Chébli-Triméche hypergroups.Mathematics Subject Classification 2000: 46F12     

The Schwartz Space and Homogeneous Distributions on H-Type Groups

 Let N be an H-type group of homogeneous dimension Q. We study the space of biradial Schwartz functions on N by means of the Gelfand transform. This enables us to characterize the class of biradial homogeneous distributions on N of degree α, with 0 ? α< Q, which are away from the identity, via the Gelfand transform. (Received 26 April 2000; in revised form December 2000)     

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。     

Quasiasymptotic expansion and the laplace transformation

The notion of quasiasymptotic expansion of distributions from S′₊ is investigated and applications on the Laplace transformation is given. Specially, an analogue of Watson lemma is given     

On a theorem of Laurent Schwartz

We give a proof of a theorem of Schwartz on Borel graphs for linear transforms between Banach spaces, completely different from the original one.     

Quasiasymptotic analysis in Colombeau algebra

We consider the main notions of G-quasiasymptotic (G-q.a.) analysis: G-q.a. behavior at zero, G-q.a. expansion at infinity, Sc-asymptotic and boundedness, Sc-expansion at infinity and their applications to the asymptotic behavior of the solution to some types of nonlinear PDEs in Colombeau algebra of generalized functions G.     

On the Schwartz space of the basic affine space

Let G be the group of points of a split reductive algebraic group G over a local field k and let X = G / U where U is the group of k-points of a maximal unipotent subgroup of G. In this paper we construct a certain canonical G-invariant space (called the Schwartz space of X) of functions on X, which is an extension of the space of smooth compactly supported functions on X. We show that the space of all elements of , which are invariant under the Iwahori subgroup I of G, coincides with the space generated by the elements of the so called periodic Lusztig basis, introduced recently by G. Lusztig (cf. [10] and [11]). We also give an interpretation of this space in terms of a certain equivariant K-group (this was also done by G. Lusztig — cf. [12]). Finally we present a global analogue of , which allows us to give a somewhat non-traditional treatment of the theory of the principal Eisenstein series.     

On the Fourier Transform of Causal Distributions

We extend the distributional Bochner formula [1, p. 72, Theorem 26] to certain kinds of distributions. Theorem I.1 gives a formula [Eq. (I.1.14)] which makes it possible to obtain easily the Fourier transform of distributions of the form As applications of the formula (I.1.14) we evaluate the Fourier transforms of the distributions G_α(P±i0, m, n) [Eq. (I.4.1)] and H_α(P±i0,n) [Eq. (II.1.1)]. It follows from Theorem II.3 that H_zk(P±i0,n) is, for 2K≠n+2r, r=0,1..., an elementary solution of the n-dimensional ultrahyperbolic operator iterated k times.     

On the Parametrization of the Afocal Stable Distributions

Thanks to Hall's documentation of a situation he described asa ‘Comedy of errors’ in the earlier literature onstable distributions, there is now general agreement as to theparametrization of these distributions when standardized asto location and scale. However, two subtly different parametrizationsof location and scale remain current in the afocal stable cases,defined as those with characteristic exponent = 1 and skewnessparameter ß 0. The more widely used parametrization,adopted by Hall and two recent monographs, lacks the linearityproperty that is ordinarily expected of location and scale parameters.This note shows that the alternative parametrization has thislinearity property, and compares the implications of the twoparametrizations for properties of stable distributions. A thirdrecent monograph seeks to avoid these issues by denying thatthe afocal stable distributions are stable at all, but it isshown that they are, in fact, integral members of the stablefamily.     

On Hyperfinite Representations of Distributions

It is shown that an internal step function on an s-dense ‘timeline’ approximating a Schwartz distribution is not necessarilyinvariant under internal infinitesimal permutations of the arguments.This invariance holds exactly if the step function is near standardas a signed -additive Radon measure. Present address: Wässigertal 27, D-5480 Remagen, W. Germany     

On the Schwartz space isomorphism theorem for rank one symmetric space

In this paper we give a simpler proof of the L ^p -Schwartz space isomorphism (0 < p ≤ 2) under the Fourier transform for the class of functions of left δ-type on a Riemannian symmetric space of rank one. Our treatment rests on Anker’s [2] proof of the corresponding result in the case of left K-invariant functions on X. Thus we give a proof which relies only on the Paley-Wiener theorem.     

On Asymptotic Proximity of Distributions

We consider some general facts concerning the convergence