We consider ordered spaces of continuous vector-valued functions on a locally compact Hausdorff space, endowed with appropriate locally convex topologies. Using suitable sets of such functions as test systems a Korovk...
详细信息
We consider ordered spaces of continuous vector-valued functions on a locally compact Hausdorff space, endowed with appropriate locally convex topologies. Using suitable sets of such functions as test systems a Korovkin type approximation theorem for equicontinuous nets of positive operators is established. As in the classical theory, the Korovkin closure is characterized both through envelopes of functions and through measure theoretical conditions. (C) 2013 Elsevier Inc. All rights reserved.
We introduce a new class FV(Omega, E) of weighted spaces of functions on a set Omega with values in a locally convex Hausdorff space E which covers many classical spaces of vector-valued functions like continuous, smo...
详细信息
We introduce a new class FV(Omega, E) of weighted spaces of functions on a set Omega with values in a locally convex Hausdorff space E which covers many classical spaces of vector-valued functions like continuous, smooth, holomorphic or harmonic functions. Then we exploit the construction of FV(Omega, E) to derive sufficient conditions such that FV(Omega, E) can be linearised, i.e. that FV(Omega, E) is topologically isomorphic to the epsilon-product FV(Omega)epsilon E where FV(Omega) := FV(Omega, K) and K is the scalar field of E.
The Banach space is retopologized by , , where is the norm in the given Banach space . It is shown here that this topology coincides with the usual weak topology of on a wide class of weakly compact subsets.
The Banach space is retopologized by , , where is the norm in the given Banach space . It is shown here that this topology coincides with the usual weak topology of on a wide class of weakly compact subsets.
This paper provides some first steps in developing empirical process theory for functions taking values in a vector space. Our main results provide bounds on the entropy of classes of smooth functions taking values in...
详细信息
This paper provides some first steps in developing empirical process theory for functions taking values in a vector space. Our main results provide bounds on the entropy of classes of smooth functions taking values in a Hilbert space, by leveraging theory from differential calculus of vector-valued functions and fractal dimension theory of metric spaces. We demonstrate how these entropy bounds can be used to show the uniform law of large numbers and asymptotic equicontinuity of the function classes, and also apply it to statistical learning theory in which the output space is a Hilbert space. We conclude with a discussion on the extension of Rademacher complexities to vector-valued function
This paper aims to present a state-of-the-art review of recent development within the areas of dynamic programming and optimal control for vector-valued functions.
This paper aims to present a state-of-the-art review of recent development within the areas of dynamic programming and optimal control for vector-valued functions.
In this paper, different types of saddle pairs of vector-valued functions are investigated. Main properties of these pairs are examined. Structure of images of saddle pair sets is found and these images are constructe...
详细信息
In this paper, different types of saddle pairs of vector-valued functions are investigated. Main properties of these pairs are examined. Structure of images of saddle pair sets is found and these images are constructed by means of cone extreme points in an explicit way as well. The obtained results provide possibility to construct dual problems in general cases of multiple objective problems and to investigate how to solve them using saddle pairs approach.
We are concerned here with Sobolev-type spaces of vector-valued functions. For an open subset Omega subset of R-N and a Banach space V, we characterize the functions in the Sobolev-Reshetnyak space R-1,R-p (Omega, V),...
详细信息
We are concerned here with Sobolev-type spaces of vector-valued functions. For an open subset Omega subset of R-N and a Banach space V, we characterize the functions in the Sobolev-Reshetnyak space R-1,R-p (Omega, V), where 1 <= p <= infinity, in terms of the existence of partial metric derivatives or partial w*-derivatives with suitable integrability properties. In the case p = infinity the Sobolev-Reshetnyak space R-1,R-infinity (Omega, V) is characterized in terms of a uniform local Lipschitz property. We also consider the special case of the space V = l(infinity). (C) 2022 The Authors. Published by Elsevier Inc.
We obtain several Banach-Stone type theorems for vector-valued functions in this paper. Let X, Y be realcompact or metric spaces, E, F locally convex spaces, and phi a bijective linear map from C(X, E) onto C(Y, F). I...
详细信息
We obtain several Banach-Stone type theorems for vector-valued functions in this paper. Let X, Y be realcompact or metric spaces, E, F locally convex spaces, and phi a bijective linear map from C(X, E) onto C(Y, F). If phi preserves zero set containments, i.e., z(f) subset of z(g) <-> z(phi(f)) subset of z(phi(g)), for all f, g is an element of C(X, E), then X is homeomorphic to V. and phi is a weighted composition operator. The above conclusion also holds if we assume a seemingly weaker condition that phi preserves nonvanishing functions, i.e., z(f) = empty set <-> z(phi f) = empty set, for all f is an element of C(X, E). These two results are special cases of the theorems in a very general setting in this paper, covering bounded continuous vector-valued functions on general completely regular spaces, and uniformly continuous vector-valued functions on metric spaces. Our results extend and generalize many recent ones. Crown Copyright (C) 2012 Published by Elsevier Inc. All rights reserved.
In this paper, we prove some basic results concerning the best approximation of vector-valued functions by algebraic and trigonometric polynomials with coefficients in normed spaces, called generalized polynomials. Th...
详细信息
In this paper, we prove some basic results concerning the best approximation of vector-valued functions by algebraic and trigonometric polynomials with coefficients in normed spaces, called generalized polynomials. Thus we obtain direct and inverse theorems for the best approximation by generalized polynomials and results concerning the existence (and uniqueness) of best approximation generalized polynomials.
In this paper we consider the norm of composition operators on vector-valued function spaces. Let X be a complex Banach space and theta be an inner function. We show that the norm of the composition operator C-theta o...
详细信息
In this paper we consider the norm of composition operators on vector-valued function spaces. Let X be a complex Banach space and theta be an inner function. We show that the norm of the composition operator C-theta on the vector-valued Hardy space H-p(T,X) (1 <= p vector-valued function on the unit circle T and showed some properties in composition operators on vector-valued function spaces, and operator-valued function spaces.
暂无评论