Analytic Feynman Integrals of Functionals in a Banach Algebra Involving the First Variation

This paper deals with the analytic Feynman integral of functionals on a Wiener space. First the authors establish the existence of the analytic Feynman integrals of functionals in a Banach algebra S_α. The authors then obtain a formula for the first variation of integrals. Finally, various analytic Feynman integration formulas involving the first variation are established.     

Fourier-Feynman transform, convolution and first variation

We examine several interesting relationships and expressions involving Fourier-Feynman transform, convolution product and first variation for functionals in the Fresnel class F(B) of an abstract Wiener space B. We also prove a translation theorem and Parseval's identity for the analytic Feynman integral. This revised version was published online in June 2006 with corrections to the Cover Date.     

On Stein?s method for infinite-dimensional Gaussian approximation in abstract Wiener spaces

In this paper, we generalize Stein?s method to “infinite-variate” normal approximation that is an infinite-dimensional approximation by abstract Wiener measures on a real separable Banach space. We first establish a Stein?s identity for abstract Wiener measures and solve the corresponding Stein?s equation. Then we will present a Gaussian approximation theorem using exchangeable pairs in an infinite-variate context. As an application, we will derive an explicit error bound of Gaussian approximation to the distribution of a sum of independent and identically distributed Banach space-valued random variables based on a Lindeberg-Lévy type limit theorem. In addition, an analogous of Berry-Esséen type estimate for abstract Wiener measures will be obtained.     

Generalized Fourier–Feynman transforms and generalized convolution products on Wiener space

In this paper we define a more general convolution product of functionals on Wiener space and develop the fundamental relationships between the generalized Fourier–Feynman transform and the convolution product.     

Inclusions of Waterman–Shiba spaces into generalized Wiener classes

The characterization of the inclusion of Waterman–Shiba spaces ΛBV^(p)$\Lambda {\mathit{BV}}^{(p)}$ into generalized Wiener classes of functions BV(q;δ)$\mathit{BV}(q;\delta )$ is given. It uses a new and shorter proof and extends an earlier result of U. Goginava.     

The Skorohod integral in conuclear spaces

We define an anticipative stochastic integral with respect to a nonhomogeneous Wiener process in a dual of a nuclear space and investigate its basic properties. The theory is developed without the use of chaos expansions.     

The Atiyah class of generalized holomorphic vector bundles

We introduce the notion of Atiyah class of a generalized holomorphic vector bundle, which captures the obstruction to the existence of generalized holomorphic connections on the bundle. As in the classical holomorphic case, this Atiyah class can be defined in three different ways: using Čech cohomology, using the first-jet short exact sequence, or adopting the Lie pair point of view.     

The anticipative Stratonovich integral in conuclear spaces

We define an anticipative stochastic integral of Stratonovich type with respect to a nonhomogeneous Wiener process in the dual of a nuclear space and investigate its basic properties.This research was partially supported by Komitet Bada Naukowych, Grant 2 1094 91 01.     

广义n参数Wiener过程和广义OUPn的导出过程的注记

记为非负单增函数且一阶偏导数均存．本文证明：（ｉ）广义ｎ参数Ｗｉｅｎｅｒ过程（ＧＢｎ）和广义ｎ参数Ｏｒｎｓｔｅｉｎ－Ｕｈｌｅｎｂｅｃｋ过程（广义ＯＵＰｎ）在Ｄ中导出的过程是马氏过程；（ｉｉ）如果ＧＢｎ（或广义ＯＵＰｎ）在Ｄ中导出的过程成为某个ＧＢｋ（或某个广义ＯＵＰｋ），则ｆｊ（ｔ１…，ｔｋ）未必恒为正常数．（ｊ＝ｋ＋１，…，ｎ）．     

Schatten class Toeplitz operators on generalized Fock spaces

In this paper we characterize the Schatten p   class membership of Toeplitz operators with positive measure symbols acting on generalized Fock spaces for the full range 0<p<∞$0<p<\infty$.     

Boundedness of generalized higher commutators of Marcinkiewicz integrals

Let (b) ＝ (b1,…,bm) be a finite family of locally integrable functions. Then,we introduce generalized higher commutator of Marcinkiwicz integral as follows:μ(b)Ω=(∫∞o|F(b)Ω,t(f)(x)|2et/t)1/2,whereF(b)Ω(f)(x)＝1/t∫|x-y|≤tΩ(x-y)/|x-y|n-1m∏j=1(bj(x)-bj(y))f(y)dy.When bj ∈(A)βj, 1≤j≤m, 0＜βj＜1,m∑j=1βj =β＜n, and Ω is homogeneous of degree zero and satisfies the cancelation condition, we prove that μ(b)Ω is bounded from Lp(Rn)to Ls(Rn), where 1 ＜ p ＜ n/β and 1/s = 1/p -β/n. Moreover, if Ω also satisfies some Lq-Dini condition, then μ(b)Ω is bounded from Lp(Rn) to (F)β,∞p(Rn) and on certain Hardy spaces. The article extends some known results.     

Notes on Riesz''s theorem on fuzzy measure space

Riesz's theorems on fuzzy measure space have been improved in essence.     

A class of infinite dimensional stochastic processes with unbounded diffusion

The paper studies Dirichlet forms on the classical Wiener space and the Wiener space over non-compact complete Riemannian manifolds. The diffusion operator is almost everywhere an unbounded operator on the Cameron–Martin space. In particular, it is shown that under a class of changes of the reference measure, quasi-regularity of the form is preserved. We also show that under these changes of the reference measure, derivative and divergence are closable with certain closable inverses. We first treat the case of the classical Wiener space and then we transfer the results to the Wiener space over a Riemannian manifold.     

Notes on elation generalized quadrangles

Let be a finite generalized quadrangle of order (s,t),s,t>1. An “elation about a point p” of is an automorphism fixing p linewise and fixing no point which is not collinear with p. An elation that generates a cyclic group of elations is called a “standard elation”. One of the problems already considered in Payne and Thas (Finite Generalized Quadrangles (1984)) is to determine just when the set of elations about the point (∞) is a group. The purpose of this paper is to provide an example where this is not the case, and then to show that for a flock generalized quadrangle the usual group of elations about (∞) is the complete set of standard elations about (∞).     

Integration by parts formulas involving generalized Fourier-Feynman transforms on function space

In an upcoming paper, Chang and Skoug used a generalized Brownian motion process to define a generalized analytic Feynman integral and a generalized analytic Fourier-Feynman transform. In this paper we establish several integration by parts formulas involving generalized Feynman integrals, generalized Fourier-Feynman transforms, and the first variation of functionals of the form where denotes the Paley-Wiener-Zygmund stochastic integral .

     

Fresnel Tomography: A Novel Approach to Wave-Function Reconstruction Based on the Fresnel Representation of Tomograms

We introduce a new type of tomographic probability distribution containing complete information about the density matrix (wave function) related to the Fresnel transform of the complex wave function. We elucidate the relation to the symplectic tomographic probability distribution. We present a multimode generalization of the Fresnel tomography and give examples of applications of the present approach. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 2, pp. 384–393, August, 2005.     

在R^d的VC类上Wiener测度的上界

本文考察指标集为d维欧氏空间Rd的子集类c的Wiener过程W的样本轨道性质．在某些情形中，对0＜an≤nd，an是不减的且an／nd是不增的假设下，求得函数Ψn使给出了Wiener测度在Rd的VC类的界，它也推广了Csorgo－Revesz关于Wiener过程增量的结果于这一一般情形．     

NOTES ON GENERALIZED EXPONENTIAL DICHOTOMIES

ＮＯＴＥＳＯＮＧＥＮＥＲＡＬＩＺＥＤＥＸＰＯＮＥＮＴＩＡＬＤＩＣＨＯＴＯＭＩＥＳ￥ＺＨＡＮＧＷＥＩＮＩＡＮ（张伟年）（ＣｅｎｔｅｒｆｏｆＭａｔｈｅｍａｔｉｃａｌＳｃｉｅｎｃｅｓＣＩＣＡ，ｔｈｅＣｈｉｎｅｓｅＡｃａｄｅｍｙｏｆＳｃｉｅｎｃｅｓ，Ｃｈｅｎ...     

Notes on Jung-Kim-Srivastava integral operator

The object of the present paper is to investigate some properties of certain integral operator Q_β^α introduced and studied recently by Jung, Kim, and Srivastava [J. Math. Anal. Appl. 176 (1993) 138-147].