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。     

Balanced colombeau products of the distributions x±-p and x-p

Results on singular products of the distributions x _± ^-p and x ^-p for natural p are derived, when the products are balanced so that their sum exists in the distribution space. These results follow the pattern of a known distributional product published by Jan Mikusiski in 1966. The results are obtained in the Colombeau algebra of generalized functions, which is the most relevant algebraic construction for tackling nonlinear problems of Schwartz distributions.     

Surjectivity of differential operators and the division problem in certain function and distribution spaces

We give an overview on surjectivity conditions for partial differential operators and operators defined by multiplication with polynomials on certain function and distribution spaces of Laurent Schwartz. We complement the classical results by treating the surjectivity of operators on the space of slowly increasing functions and on the space of rapidly decreasing distributions, respectively.     

Generalized Pompeiu equation in distributions

We consider a generalized Pompeiu equation in the space of Schwartz distributions and as an application we find the locally integrable solutions of the equation.     

Full algebra of generalized functions and non-standard asymptotic analysis

We construct an algebra of generalized functions endowed with a canonical embedding of the space of Schwartz distributions.We offer a solution to the problem of multiplication of Schwartz distributions similar to but different from Colombeau’s solution.We show that the set of scalars of our algebra is an algebraically closed field unlike its counterpart in Colombeau theory, which is a ring with zero divisors. We prove a Hahn–Banach extension principle which does not hold in Colombeau theory. We establish a connection between our theory with non-standard analysis and thus answer, although indirectly, a question raised by Colombeau. This article provides a bridge between Colombeau theory of generalized functions and non-standard analysis.     

A Method for Solving the Hopf Equation and Generalization of Sobolev–Schwartz Distributions

We introduce distributions that are functionals with values in a nonarchimedian field of Laurent series. These objects naturally generalize Sobolev–Schwartz distributions, and we use them to find generalized solutions of the Hopf equation in the form of infinitely narrow solitons and shock waves. We propose a method for computation of the profile of an infinitely narrow soliton and of a shock wave described by the Hopf equation.     

The Plancherel decomposition for a reductive symmetric space. II. Representation theory

We obtain the Plancherel decomposition for a reductive symmetric space in the sense of representation theory. Our starting point is the Plancherel formula for spherical Schwartz functions, obtained in part I. The formula for Schwartz functions involves Eisenstein integrals obtained by a residual calculus. In the present paper we identify these integrals as matrix coefficients of the generalized principal series.     

Stability of exponential equations in Schwartz distributions

We reformulate the superstability of exponential equation and cosine functional equation [J.A. Baker, The stability of cosine equation, Proc. Amer. Math. Soc. 80 (1980) 411–416] in some spaces of generalized functions such as the Schwartz distributions, Sato hyperfunctions, and Gelfand generalized functions, which completes the previous results of partial generalizations of the stability problems [J. Chung, A distributional version of functional equations and their stabilities, Nonlinear Anal. 62 (2005) 1037–1051; J. Chung, S.Y. Chung, D. Kim, The stability of Cauchy equations in the space of Schwartz distributions, J. Math. Anal. Appl. 295 (2004) 107–114].     

Existence and uniqueness of v-asymptotic expansions and Colombeau's generalized numbers

We define a type of generalized asymptotic series called v-asymptotic. We show that every function with moderate growth at infinity has a v-asymptotic expansion. We also describe the set of v-asymptotic series, where a given function with moderate growth has a unique v-asymptotic expansion. As an application to random matrix theory we calculate the coefficients and establish the uniqueness of the v-asymptotic expansion of an integral with a large parameter. As another application (with significance in the non-linear theory of generalized functions) we show that every Colombeau's generalized number has a v-asymptotic expansion. A similar result follows for Colombeau's generalized functions, in particular, for all Schwartz distributions.     

广义数及其应用(Ⅰ)

本文在文献[2]的基础上引进广义数系统,定义了以广义数为基础的广义函数(本质不同于L.Schwartz的分布),研究了勒贝格积分的推广,将这理论应用于分布,便得到对σ函数等的自然理解,对广义数应用于量子场论中,也作了一些尝试性的工作。     

广义函数空间上Drygas型函数方程的稳定性

在广义函数空间上重新定义了Drygas型函数方程的稳定性,然后利用高斯变换将广义函数空间上的函数方程稳定性转换为R~(n+1)空间上的光滑函数方程的稳定性.在求得正则化的函数方程稳定性后,利用广义函数与正则化函数间的关系给出在广义函数空间上的Drygas型函数方程的Hyers-Ulam-Rassias型稳定性.     

Conditional expectations and submartingale sequences of random Schwartz distributions

Let (Ω, Σ, P) be a fixed complete probability space, $\text{D}$ the real Schwartz space, and $\text{D′}$ its strong dual. $\text{D}$ and $\text{D′}$ are partially ordered by $\text{C}$ and $\text{C′}$ respectively, where $\text{C}$ is the positive cone of nonnegative functions in $\text{D}$ and $\text{C′}$ its dual in $\text{D′}$. $\text{C}$ is a strict $\text{B}$-cone and $\text{C′}$ is normal, where $\text{B}$ is the family of all bounded subsets of $\text{D}$. If X, Y are two random Schwartz distributions, then X ≤ Y if and only if Y(ω) ? X(ω) ∈ $\text{D′}$ for almost all $\text{ω ∈ Ω(P)}$. Integrability of random Schwartz distributions and properties of such integrals are discussed. The monotone convergence theorem, the dominated convergence theorem, and Fatou's lemma are proved. The existence of conditional expectations of integrable random Schwartz distributions relative to a given sub σ-field of Σ is shown. Properties of conditional expectations are discussed and the conditional form of the monotone convergence theorem is proved. Sub(super)-martingale sequences are defined via the partial order relations introduced above, and a convergence theorem is given. The notion of a potential is introduced and the Riesz decomposition theorem is proved.     

Bochner Schwartz Type Theorem for Conditionally Positive Definite Fourier Hyperfunctions*

We prove the Bochner–Schwartz type theorem for conditionally positive definite Fourier hyperfunctions which generalizes the result of Gelfand-Vilenkin in their treatise Generalized functions, vol. IV for distributions.     

The de-Rham theorem and Shapiro lemma for Schwartz functions on Nash manifolds

In this paper we continue our work on Schwartz functions and generalized Schwartz functions on Nash (i.e. smooth semi-algebraic) manifolds. Our first goal is to prove analogs of the de-Rham theorem for de-Rham complexes with coefficients in Schwartz functions and generalized Schwartz functions. Using that we compute the cohomologies of the Lie algebra g of an algebraic group G with coefficients in the space of generalized Schwartz sections of G-equivariant bundle over a G-transitive variety M. We do it under some assumptions on topological properties of G and M. This computation for the classical case is known as the Shapiro lemma.     

Micropolar fluid system in a space of distributions and large time behavior

We analyze the well-posedness of the initial value problem for the generalized micropolar fluid system in a space of tempered distributions and also prove the existence of the stationary solutions. The asymptotic stability of solutions is showed in this space, and as a consequence, a criterium for vanishing small perturbations of initial data (stationary solution) at large time is obtained. A fast decay of the solutions is obtained when we assume more regularity on the initial data.     

Controllability of Differential-Algebraic Equations in the Class of Impulse Effects

Considering a control linear system of differential-algebraic equations with infinitely differentiable coefficients we establish the existence of solutions in the class of Sobolev–Schwartz distributions. The solution is expressed as the sum of a regular generalized function and a singular generalized function. We study controllability with a jump of a regular component and a singular component of the solution.     

Approximately additive Schwartz distributions

Generalizing the Hyers-Ulam-Rassias stability theorem [Th.M. Rassias, On the stability of linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297-300] to the space of Schwartz distributions, we introduce a concept of approximately additive Schwartz distributions and prove that every approximately additive distribution can be approximated by linear functions.     

Spectra of Pseudo-Differential Operators on the Schwartz Space

We initiate the study of the spectrum, point spectrum, continuous spectrum and residual spectrum of a constant coefficient pseudo-differential operator on the Schwartz space. The results are in sharp contrast with the corresponding ones for the Banach space L^p(Rⁿ), 1 ≤ p ≤ ∞. An application to the study of spectra of constant coefficient pseudo-differential operators on the space of Schwartz distributions is given.     

Marshall and Olkin’s Distributions

A review is provided of the continuous and discrete distributions introduced by the eminent Professors Marshall and Olkin. The topics reviewed include: bivariate geometric distribution, extreme value behavior, bivariate negative binomial distribution, bivariate exponential distribution, concomitants, reliability, distributions of sums and ratios, Ryu’s bivariate exponential distribution, bivariate Pareto distribution and generalized exponential and Weibull distributions. Some hitherto unknown results about these distributions are also mentioned. This is a tribute to the work of Professors Marshall and Olkin.     

Distributional Methods for a Class of Functional Equations and Their Stabilities

We consider a class of n-dimensional Pompeiu equations and that of Pexider equations and their Hyers Ulam stability problems in the spaces of Schwartz distributions. First, reducing the given distribution version of functional equations to differential equations we find their solutions. Secondly, using approximate identities we prove the Hyers Ulam stability of the equations.