Uniform approximation: The non-locally convex case

IfS is a compact Hausdorff space of finite covering dimension and (E, τ) is a real or complex topological vector space (not necessarily locally convex), we prove a Weierstrass-Stone theorem for subsets ofC(S;E), the space of all continuous functions fromS intoE, equipped with the topology of uniform convergence overS.     

Hypercyclic operators on non-locally convex spaces

We transfer a number of fundamental results about hypercyclic operators on locally convex spaces (due to Ansari, Bès, Bourdon, Costakis, Feldman, and Peris) to the non-locally convex situation. This answers a problem posed by A. Peris [Multi-hypercyclic operators are hypercyclic, Math. Z. 236 (2001), 779-786].

     

The approximation on locally convex spaces

In this paper, we give some properties of f-approximation, f-Chebyshev centers and f-farthest points in locally convex spaces. Supported by NSFC     

The polya-1 property for convex approximation

If f∈L_p[0, 1], let f_p be its best L_p-approximant by convex functions. It is shown that if exists uniformly on closed subintervals of (0,1). This research was partially supported by Grant No. 020-033-58 from the Faculty Research Committee, Idaho State University.     

The convex floating body and polyhedral approximation

We consider the convex floating body of a polytope and polyhedral approximation of a convex body. Supported by NSF grant DMS-8902327.     

The simultaneous approximation in the locally convex spaces

In this paper, we study the characterization of f-Chebyshev radus and f-Chebyshev centers and the existence of f-Chebyshev centers in locally convex spaces. Research supported by the National Science Foundation of F. R. China     

Smooth approximation of convex bodies

We describe a general approximation procedure for convex bodies which shows, in particular, that a body of constant width can be approximated, in the Hausdorff metric, by bodies of constant width with analytic boundaries (in fact, with algebraic support functions). Moreover, the approximating bodies have (at least) the same symmetries as the original one.     

Uniform duality in semi-infinite convex optimization

We study infinite sets of convex functional constraints, with possibly a set constraint, under general background hypotheses which require closed functions and a closed set, but otherwise do not require a Slater point. For example, when the set constraint is not present, only the consistency of the conditions is needed. We provide hypotheses, which are necessary as well as sufficient, for the overall set of constraints to have the property that there is no gap in Lagrangean duality for every convex objective function defined on ℝⁿ. The sums considered for our Lagrangean dual are those involving only finitely many nonzero multipliers. In particular, we recover the usual sufficient condition when only finitely many functional constraints are present. We show that a certain compactness condition in function space plays the role of finiteness, when there are an infinite number of functional constraints. The author's research has been partially supported by Grant ECS8001763 of the National Science Foundation.     

Uniform approximation of vector-valued functions

Summary Uniform approximation of vector-valued functions is defined, together with analogs of extreme points and H-sets. Characterizations of best approximations are given in terms of these, and some applications are presented.Most of the work for this paper was done at Michigan State University, East Lansing; as a Ph. D. Thesis directed by D. G. MOURSUND. The author wishes to thank Prof. MOURSUND for his kind assistance.