Space Of Continuous Real Functions Research Articles

Divergence of interpolation processes on sets of the second category

C([0, 1]) is the space of real continuous functions f(x) on [0, 1] and ω(δ) is a majorant of the modulus of continuity ω(f, δ), satisfying the condition\(\mathop {\overline {\lim } }\limits_{n \to \infty } \omega (1/n) \ln n = \infty \). A solution is given to a problem of S. B. Stechkin: for any matrix\(\mathfrak{M}\) of interpolation points there exists an f(x) ɛ c([0, 1]), ω (f, δ) = o{ω(δ)} whose Lagrange interpolation process diverges on a set ℰ of second category on [0, 1].

Statistical mechanics on a compact set with $Z^\nu $ action satisfying expansiveness and specification

We consider a compact set Q with a homeomorphism (or more generally a Z' action) such that expansiveness and Bowen's specification condition hold. The entropy is a function on invariant probability measures. The pressure (a concept borrowed from statistical mechanics) is defined as function on ??(s)—the real continuous functions on ft. The entropy and pressure are shown to be dual in a certain sense, and this duality is investigated. 0. Introduction. Invariant measures for an Anosov diffeomorphism have been studied by Sinai (16), (17). More generally, Bowen (2), (3) has considered invariant measures on basic sets for an Axiom A diffeomorphism. The problems encoun- tered are strongly reminiscent of those of statistical mechanics (for a classical lattice system—see (14, Chapter 7)). In fact Sinai (18) has explicitly used techniques of statistical mechanics to show that an Anosov diffeomorphism does not in general have a smooth invariant measure. In this paper, we rewrite a part of the general theory of statistical mechanics for the case of a compact set s satisfying expansiveness and the specification property of Bowen (2). Instead of a Z action we consider a Z' action as is usual in lattice statistical mechanics, where Q = Fz' (F: a finite set). This rewriting gives a more general and intrinsic formulation of (part of) statistical mechanics; it presents a number of technical problems, but the basic ideas are contained in the papers of Gallavotti, Lanford, Miracle, Robinson, and Ruelle (7), (11), (12), (13), etc. The ideas of Bowen (2) and Goodwyn (8) on the relation between topological and measure-theoretical entropy are also used. We describe now some of our results in the case of a homeomorphism F of a metrizable compact set n satisfying expansiveness and specification (see §1). Let na = {x G n: Tx = {x}}, and let <3(s) be the Banach space of real continuous functions on n. The pressure F is a continuous convex function on u(fi) defined by

Corrections to: “On sequential convergence”

In my paper [2], Theorems 8.2 and 8.3 are false. I am indebted to Dennis Sentilles who sent me a counterexample to Theorem 8.2. The wrong statements concern LS-spaces. These are linear spaces S on which there is a metric p and a function f such that xn -x in S if and only if both p(xn, x) O-0 and f(xn) remains bounded. Without restating the further conditions imposed on p and f, we single out for the present two tractable subclasses of the class of LS-spaces: (I) LF-spaces, i.e. strict inductive limits of spaces ([1], [2, Theorem 7.7]). (II) If B is a separable Banach space, and S is its dual B* with weak-star convergence of sequences, then S is an LS-space [2, Theorem 7.1]. (Actually the same is true with Frechet in place of Banach: see [1, Theoreme 5, p. 84].) In case (I), Theorems 8.2 and 8.3 are true. The Banach-Steinhaus theorem for LF-spaces is well known [1, Theoreme 2, p. 73]. But in case (II), if B is infinitedimensional, the elements of its unit ball give pointwise bounded continuous linear forms on S which are not equicontinuous for the LS-topology. The proof of Theorem 8.2 in [2] errs in assuming that multiplication by a positive scalar is an open mapping in the relative topology of a (convex, symmetric, closed, metrizable) set. The proof of Theorem 8.3 is invalid since it rests on Theorem 8.2. Disproving the statement of 8.3 requires some further work. Here is one counterexample. Let ' be the space of continuous real functions on [0, 1] with supremum norm. Then its dual ' * is the space of finite signed measures on [0, 1]. W* with weak* sequential convergence is an LS-space. It has a countable dense set and is complete as an LS-space. The LS-topology on W* is the topology of uniform convergence on sequences {f,} in ' with f,11 ---> 0 [2, around Theorem 7.8]. By the MackeyArens theorem, the dual space of (V*, Y) is ' (see also [1, Theoreme 6, p. 85]). A sequence in W is weakly convergent if and only if it is uniformly bounded and converges pointwise. Thus the following fact contradicts Theorem 8.3:

A Choquet-Deny theorem for affine functions on a Choquet simplex

The closed wedges in C(X) (the space of real continuous functions on a compact Hausdorff space X) which are also inf-lattices have been characterized by Choquet and Deny (2); see also (5). The present note extends their result to certain wedges of affine continuous functions on a Choquet simplex, the generalization being in the same spirit as the generalization of the Kakutani- Stone theorem obtained by Edwards in (4).I should like to thank my supervisor, Dr D. A. Edwards, for suggesting this problem and for his subsequent help. I am also grateful to the referee for correcting several slips.

The Wiener Measure of Hilbert Neighborhoods in the Space of Real Continuous Functions

The Wiener Measure of Hilbert Neighborhoods in the Space of Real Continuous Functions

Metric spaces and positive definite functions

As poo we get the space Em with the distance function maxi-, ... I xi X. Let, furthermore, lP stand for the space of real sequences with the series of pth powers of the absolute values convergent. Similarly let LP denote the space of real measurable functions in the interval (0, 1) which are summable to the pth power, while C shall mean the space of real continuous functions in the same interval. In all these spaces a distance function is assumed to be defined as usual. t L2 is equivalent to the real Hilbert space t. The spaces EmP, IP and LP are metric only if p > 1, but we shall consider them also for positive values of p O). A general theorem of Banach and Mazur ([1], p. 187) states that any separable metric space (5 may be imbedded isometrically in the space C. Furthermore, as a special case of a well known theorem of Urysohn, any such space (E may be imbedded topologically in t. Isometric imbeddability of (E in '& is, however, a much more restricted property of (B. The chief purpose of this paper is to point out the intimate relationship between the problem of isometric imbedding and the concept of positive definite functions, if this concept is properly enlarged. As a first approach to this connection we consider here isometric imbedding in Hilbert space only. It turns out that the possibility of imbedding$ in 6& is very easily expressible in terms of the elementary function e-t2 and the concept of positive definite functions (Theorem 1). The author's previous result ([10]) to the effect that i(,y), (O <,y < 1), which is the space arising from 6& by raising its metric to a