Unitary representations of the groups of measurable and continuous functions with values in the circle

We give a classification of unitary representations of certain Polish, not necessarily locally compact, groups: the groups of all measurable functions with values in the circle and the groups of all continuous functions on compact, second countable, zero-dimensional spaces with values in the circle. In the proofs of our classification results, certain structure theorems and factorization theorems for linear operators are used.     

Operators Represented by Conditional Expectations and Random Measures

On standard measure spaces every order continuous linear map between two ideals of almost everywhere finite measurable functions can be represented by a random measure. An analogue of this theorem is proved for the case of arbitrary σ-finite measure spaces. This fact leads to a proof that every order continuous linear map between ideals of almost everywhere finite measurable functions on σ-finite measure spaces is multiplication conditional expectation representable. This sheds further light on the structure of order continuous operators. Mathematics Subject Classification (20000): 47B38, 47B65 J.J. Grobler-Financial assistance of the NRF is gratefully acknowledged. D.T. Rambane-Financial assistance of the University of Venda research fund is gratefully acknowledged.     

The Cauchy Transform of Continuous Linear Functionals and Projections on the Weighted Spaces of Analytic Functions

Under some general assumptions on weight, we give a complete characterization of the Cauchy transform of continuous linear functionals on the weighted spaces of holomorphic functions in a ball. We construct an integral projection that sends the weighted spaces of measurable and n-harmonic functions in the ball onto the corresponding spaces of holomorphic functions.     

Virtual continuity of measurable functions of several variables and embedding theorems

Luzin’s classical theorem states that any measurable function of one variable is “almost” continuous. This is no longer true for measurable functions of several variables. The search for a correct analogue of Luzin’s theorem leads to the notion of virtually continuous functions of several variables. This, probably new, notion appears implicitly in statements such as embedding theorems and trace theorems for Sobolev spaces. In fact, it reveals their nature of being theorems about virtual continuity. This notion is especially useful for the study and classification of measurable functions, as well as in some questions on dynamical systems, polymorphisms, and bistochastic measures. In this work we recall the necessary definitions and properties of admissible metrics, define virtual continuity, and describe some of its applications. A detailed analysis will be presented elsewhere.     

A differential operator and weak topology for Lipschitz maps

We show that the Scott topology induces a topology for real-valued Lipschitz maps on Banach spaces which we call the L-topology. It is the weakest topology with respect to which the L-derivative operator, as a second order functional which maps the space of Lipschitz functions into the function space of non-empty weak^∗ compact and convex valued maps equipped with the Scott topology, is continuous. For finite dimensional Euclidean spaces, where the L-derivative and the Clarke gradient coincide, we provide a simple characterization of the basic open subsets of the L-topology. We use this to verify that the L-topology is strictly coarser than the well-known Lipschitz norm topology. A complete metric on Lipschitz maps is constructed that is induced by the Hausdorff distance, providing a topology that is strictly finer than the L-topology but strictly coarser than the Lipschitz norm topology. We then develop a fundamental theorem of calculus of second order in finite dimensions showing that the continuous integral operator from the continuous Scott domain of non-empty convex and compact valued functions to the continuous Scott domain of ties is inverse to the continuous operator induced by the L-derivative. We finally show that in dimension one the L-derivative operator is a computable functional.     

Nonautonomous Exponential Dichotomy

In this note we generalize the strongly continuous bisemigroups generated by exponentially dichotomous operators to so-called bievolution families. These families are then related to strongly continuous bisemigroups on certain Banach spaces of continuous and measurable vector-valued functions. The research leading to this note was supported by the Italian Ministry of Education and Research (MIUR) under PRIN grant no. 2006017542-003, and by INdAM-GNCS.     

Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains

Summary. This paper is devoted to the study of the finite volume methods used in the discretization of conservation laws defined on bounded domains. General assumptions are made on the data: the initial condition and the boundary condition are supposed to be measurable bounded functions. Using a generalized notion of solution to the continuous problem (namely the notion of entropy process solution, see [9]) and a uniqueness result on this solution, we prove that the numerical solution converges to the entropy weak solution of the continuous problem in for every . This also yields a new proof of the existence of an entropy weak solution. Received May 18, 2000 / Revised version received November 21, 2000 / Published online June 7, 2001     

Hausdorff Dimension of the Set of Generic Points for Gibbs Measures

For a Gibbs measure on the configuration space of a finite spin lattice system, we find (in terms of entropy) the Hausdorff dimension of the set of generic points. Using this result, we evaluate the Hausdorff dimension of level sets for Birkhoff ergodic averages of some continuous functions on the configuration space.     

On a Generalization of Szász–Mirakjan–Kantorovich Operators

In this paper we introduce and study a sequence of positive linear operators acting on suitable spaces of measurable functions on [0,+∞[, including L ^p ([0,+∞[) spaces, 1 ≤ p < +∞, as well as continuous function spaces with polynomial weights. These operators generalize the Szász–Mirakjan–Kantorovich operators and they allow to approximate (or to reconstruct) suitable measurable functions by knowing their mean values on a sequence of subintervals of [0,+∞[ that do not constitute a subdivision of it. We also give some estimates of the rates of convergence by means of suitable moduli of smoothness.     

Differentiability of p-Harmonic Functions on Metric Measure Spaces

We study p-harmonic functions on metric measure spaces, which are formulated as minimizers to certain energy functionals. For spaces supporting a p-Poincaré inequality, we show that such functions satisfy an infinitesmal Lipschitz condition almost everywhere. This result is essentially sharp, since there are examples of metric spaces and p-harmonic functions that fail to be locally Lipschitz continuous on them. As a consequence of our main theorem, we show that p-harmonic functions also satisfy a generalized differentiability property almost everywhere, in the sense of Cheeger’s measurable differentiable structures.     

On the approximation of measurable functions by continuous functions

We present a characterization of the completed Borel measure spaces for which every measurable function, with values in a separable Frechet space, is the almost everywhere limit of a sequence of continuous functions. From this characterization one can easily obtain results that have appeared recently in the literature, in a more general form. We also examine what happens when the range is a subset of an arbitrary Banach space, and show that this case often reduces to the separable case.     

Factoring multi-sublinear maps

Using Rademacher type, maximal estimates are established for k-sublinear operators with values in the space of measurable functions. Maurey–Nikishin factorization implies that such operators factor through a weak-type Lebesgue space. This extends known results for sublinear operators and improves some results for bilinear operators. For example, any continuous bilinear operator from a product of type 2 spaces into the space of measurable functions factors through a Banach space. Also included are applications for multilinear translation invariant operators.     

Measurability implies continuity for solutions of functional equations - even with few variables

Summary. We prove that - under certain conditions - measurable solutions $f$ of the functional equation $f(x)=h(x,y,f(g_{1}(x,y)),\ldots,f(g_{n}(x,y))),\quad(x,y)\in D \subset \mathbb{R}^{s} \times \mathbb{R}^{l}$ are continuous, even if $1\le l\le s$. As a tool we introduce new classes of functions which - roughly speaking - interpolate between continuous and Lebesgue measurable functions. Connection between these classes are also investigated.     

Topological entropy of continuous functions on topological spaces

Adler, Konheim and McAndrew introduced the concept of topological entropy of a continuous mapping for compact dynamical systems. Bowen generalized the concept to non-compact metric spaces, but Walters indicated that Bowen’s entropy is metric-dependent. We propose a new definition of topological entropy for continuous mappings on arbitrary topological spaces (compactness, metrizability, even axioms of separation not necessarily required), investigate fundamental properties of the new entropy, and compare the new entropy with the existing ones. The defined entropy generates that of Adler, Konheim and McAndrew and is metric-independent for metrizable spaces. Yet, it holds various basic properties of Adler, Konheim and McAndrew’s entropy, e.g., the entropy of a subsystem is bounded by that of the original system, topologically conjugated systems have a same entropy, the entropy of the induced hyperspace system is larger than or equal to that of the original system, and in particular this new entropy coincides with Adler, Konheim and McAndrew’s entropy for compact systems.     

Integral representation of belief measures on compact spaces

An integral representation theorem for outer continuous and inner regular belief measures on compact topological spaces is elaborated under the condition that compact sets are countable intersections of open sets (e.g. metric compact spaces). Extreme points of this set of belief measures are identified with unanimity games with compact support. Then, the Choquet integral of a real valued continuous function can be expressed as a minimum of means over the sigma-core and also as a mean of minima over the compact subsets. Similarly, for bounded measurable functions, the Choquet integral is expressed as min of means over the core, we prove in addition that it is a mean of infima over the compact subsets. Then, we obtain Choquet–Revuz' measure representation theorem and introduce the Möbius transform of a belief measure. An extension to locally compact and sigma-compact topological spaces is provided.     

Egoroff’s theorem and maximal run length

We construct a sequence of measurable functions converging at each point of the unit interval, but the set of points with any given rate of convergence has Hausdorff dimension one. This is used to show that a version of Egoroff’s theorem due to Taylor is best possible. The construction relies on an analysis of the maximal run length of ones in the dyadic expansion of real numbers. It is also proved that the exceptional set for a limit theorem of Renyi has Hausdorff dimension one.     

Max-plack-institut fiir mathematik

For certain classes of locally Lipschitzian functions (continuous selections, componentwise maximum functions of Cfc-mappings) we compute bounds on the entropy dimension of the set of generalized, critical values. This yields a measure of metric complexity to compare these classes. For vector valued functions we also compare different concepts of derivatives from this metric point of view. For continuous selections of differentiable functions even explicit bounds on the e-entropy of the nearly critical values of f are provided.     

Representations of Archimedean Riesz spaces by continuous functions

A brief survey of representations of Archimedean Riesz spaces in spaces of continuous extended real-valued functions, together with an example of their use in proving results about Riesz spaces     

Dirac operators and spectral triples for some fractal sets built on curves

We construct spectral triples and, in particular, Dirac operators, for the algebra of continuous functions on certain compact metric spaces. The triples are countable sums of triples where each summand is based on a curve in the space. Several fractals, like a finitely summable infinite tree and the Sierpinski gasket, fit naturally within our framework. In these cases, we show that our spectral triples do describe the geodesic distance and the Minkowski dimension as well as, more generally, the complex fractal dimensions of the space. Furthermore, in the case of the Sierpinski gasket, the associated Dixmier-type trace coincides with the normalized Hausdorff measure of dimension log3/log2.     

Transformations preserving the Hausdorff-Besicovitch dimension

Continuous transformations preserving the Hausdorff-Besicovitch dimension (“DP-transformations”) of every subset of R ¹ resp. [0, 1] are studied. A class of distribution functions of random variables with independent s-adic digits is analyzed. Necessary and sufficient conditions for dimension preservation under functions which are distribution functions of random variables with independent s-adic digits are found. In particular, it is proven that any strictly increasing absolutely continuous distribution function from the above class is a DP-function. Relations between the entropy of probability distributions, their Hausdorff-Besicovitch dimension and their DP-properties are discussed. Examples are given of singular distribution functions preserving the fractal dimension and of strictly increasing absolutely continuous functions which do not belong to the DP-class.