Distributions and Analytical Measures on Infinite-Dimensional Spaces

Spaces of test functions and spaces of distributions (generalized measures) on infinite-dimensional spaces are constructed, which, in the finite-dimensional case, coincide with classical spaces \(\mathscr{D}\) and \(\mathscr{D}'\). These distribution spaces contain generalized Feynman measures (but do not contain a generalized Lebesgue measure, which is not considered in this paper). For broad classes of infinite-dimensional differential equations in distribution spaces, the Cauchy problem has fundamental solutions. These results are much more definitive than those of A.Yu. Khrennikov’s and A.V. Uglanov’s pioneering works.     

Localization for Hyperbolic Measures on Infinite-Dimensional Spaces

Functional Analysis and Its Applications - Properties of the extreme points of families of concave measures on infinite-dimensional locally convex spaces are studied. The localization method is...     

Smooth Diffusion Measures and Their Transformations

In this paper, we show that some well-known equations of mathematical physics can be naturally treated from the probability viewpoint as equations for logarithmic derivatives of smooth diffusion measures. We discuss a probabilistic interpretation of the Darboux transformation applied in the theory of the SturmLiouville equations. It is shown that this transformation can be applied in deriving an explicit expression for the RadonNikodym derivative of a pair of absolutely continuous measures generated by various scalar multiplicative functionals of diffusion processes. Bibliography: 12 titles.     

无穷维欧氏空间中球体及球面的Lebesgue测度

本分别给出了使在无穷维欧氏空间中球体和球面具有有限的,但又不是无穷小的测度的半径集合。     

Smooth Lipschitz Retractions of Starlike Bodies onto their Boundaries in Infinite-Dimensional Banach Spaces

Let X be an infinite-dimensional Banach space, and let A bea C^p Lipschitz bounded starlike body (for instance the unitball of a smooth norm). We prove that:

1. the boundary A is C^p Lipschitzcontractible;
2. there is a C^p Lipschitz retraction from A ontoA;
3. there is a C^p Lipschitz map T: A A with no approximatefixedpoints.

     

Tiling Infinite-Dimensional Normed Spaces

The present paper deals with real infinite-dimensional normedspaces; some of the main concepts here make sense, and havebeen treated in the literature, in the general context of topologicalHausdorff linear spaces over reals. A subset of a normed space X is a body if it is different fromX itself and is the closure of its non-empty interior. A coveringof X by bodies is called a tiling of X whenever any two differentmembers of it have disjoint interiors. The elements of sucha covering are called tiles. A tiling is bounded (respectivelyconvex) whenever each tile is bounded (respectively convex).1991 Mathematics Subject Classification 46B20.     

Furstenberg Transformations on Cartesian Products of Infinite-Dimensional Tori

We consider in this note Furstenberg transformations on Cartesian products of infinite-dimensional tori. Under some appropriate assumptions, we show that these transformations are uniquely ergodic with respect to the Haar measure and have countable Lebesgue spectrum in a suitable subspace. These results generalise to the infinite-dimensional setting previous results of H. Furstenberg, A. Iwanik, M. Lemanzyk, D. Rudolph and the second author in the one-dimensional setting. Our proofs rely on the use of commutator methods for unitary operators and Bruhat functions on the infinite-dimensional torus.     

Markov Uniqueness and Rademacher Theorem for Smooth Measures on an Infinite-Dimensional Space under Successful-Filtration Condition

For a smooth measure on an infinite-dimensional space, a “successful-filtration” condition is introduced and the Markov uniqueness and Rademacher theorem for measures satisfying this condition are proved. Some sufficient conditions, such as the well-known Hoegh-Krohn condition, are also considered. Examples demonstrating connections between these conditions and applications to convex measures are given.__________Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 2, pp. 170–186, February, 2005.     

Local Rigidity of Infinite-Dimensional Teichmuller Spaces

This paper presents a rigidity theorem for infinite-dimensionalBergman spaces of hyperbolic Riemann surfaces, which statesthat the Bergman space A¹(M), for such a Riemann surface M,is isomorphic to the Banach space of summable sequence, l¹.This implies that whenever M and N are Riemann surfaces thatare not analytically finite, and in particular are not necessarilyhomeomorphic, then A¹(M) is isomorphic to A¹(N). It is knownfrom V. Markovic that if there is a linear isometry betweenA¹(M) and A¹(N), for two Riemann surfaces M and N of non-exceptionaltype, then this isometry is induced by a conformal mapping betweenM and N. As a corollary to this rigidity theorem presented here,taking the Banach duals of A¹(M) and l¹ shows that the spaceof holomorphic quadratic differentials on M, Q(M), is isomorphicto the Banach space of bounded sequences, l. As a consequenceof this theorem and the Bers embedding, the Teichmüllerspaces of such Riemann surfaces are locally bi-Lipschitz equivalent.     

Infinite-Dimensional Measure Spaces and Frame Analysis

We study certain infinite-dimensional probability measures in connection with frame analysis. Earlier work on frame-measures has so far focused on the case of finite-dimensional frames. We point out that there are good reasons for a sharp distinction between stochastic analysis involving frames in finite vs. infinite dimensions. For the case of infinite-dimensional Hilbert space ?, we study three cases of measures. We first show that, for ? infinite dimensional, one must resort to infinite dimensional measure spaces which properly contain ?. The three cases we consider are: (i) Gaussian frame measures, (ii) Markov path-space measures, and (iii) determinantal measures.     

Generalized Quasi-Variational Inequalities in Infinite-Dimensional Normed Spaces

In this paper, we deal with the following problem: given a real normed space E with topological dual E*, a closed convex set XE, two multifunctions :X2^X and , find such that We extend to the above problem a result established by Ricceri for the case (x)X, where in particular the multifunction is required only to satisfy the following very general assumption: each set (x) is nonempty, convex, and weakly-star compact, and for each yX–:X the set is compactly closed. Our result also gives a partial affirmative answer to a conjecture raised by Ricceri himself.     

一致光滑Banach空间中一类非线性映象的迭代过程

本文引入了广义Lipschitz的概念，研究了广义Lipschitz强增生映象的Mann型迭代和Ishikawa型迭代过程的收敛性，所得结果统一和扩展了近期相关结果．     

Partitioning Infinite-Dimensional Spaces for Generalized Riemann Integration

To form Riemann sums for generalized Riemann integrals, thedomain of integration must be partitioned in a suitable manner.The existence of the required partitions is usually proved bya simple method of repeated bisection of the domain of integration.However, when the domain is the Cartesian product of infinitelymany copies of the set of real numbers, this simple method ofproof has frequently failed. A proof which works for infinite-dimensionalspaces is provided here. 2000 Mathematics Subject Classification28C20.     

Linear Transformations on Grassmann Spaces III

We prove that a linear transformation from one grassmann space to another that takes decomposable vectors to decomposable vectors either maps the entire space into a pure subspace of the range space or is a composition of maps which are induced by linear maps and correlations between subspaces of the underlying vector spaces     

Balancing Transformations for Infinite-Dimensional Systems with Nuclear Hankel Operator

We consider balancing and model reduction by balanced truncation for infinite-dimensional linear systems. A functional analytic approach to state space transformations leading to balanced realizations is presented. These transformations can be further used to explicitly construct truncated balanced realizations. The presented approach is applicable to bounded well-posed linear systems with nuclear Hankel operator and finite-dimensional input and output space. Controllability and observability are not required.     

K—very Smooth Banach Spaces

We introduce the notion of K-very smoothness which is a generalization of very smoothness in Banach spaces. A necessary and sufficient condition for a Banach space to be K-very smooth is obtained. We also consider some relations between K-very smoothness and other geometrical notions.