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...
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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
Let (Omega,Sigma,mu) be a sigma-finite measure space, 1 Z be a bounded bilinear map. We say that an X-valued function is p-integrable with respect to B whenever sup{integral(Omega) parallel to B(f(w), y parallel to(p...
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Let (Omega,Sigma,mu) be a sigma-finite measure space, 1 <= p < infinity, X be a Banach space X and B : X x Y -> Z be a bounded bilinear map. We say that an X-valued function is p-integrable with respect to B whenever sup{integral(Omega) parallel to B(f(w), y parallel to(p) d mu : parallel to y parallel to = 1} is finite. We identify the spaces of functions integrable with respect to the bilinear maps arising from Holder's and Young's inequalities. We apply the theory to give conditions on X-valued kernels for the boundedness of integral operators TB(f)(w) = integral(Omega') B(k(w, w'), f(w'))d mu'(w') from L-p(Y) into L-p(Z), extending the results known in the operator-valued case, corresponding to B : L(X, Y) x X -> Y given by B(T, x) = Tx.
We extend the well-known Peano Kernel Theorem to a class of linear operators L C-n+1([a, b];X)--> X, X being a Banach space, which vanish on abstract polynomials of degree less than or equal to n. We then recover, ...
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We extend the well-known Peano Kernel Theorem to a class of linear operators L C-n+1([a, b];X)--> X, X being a Banach space, which vanish on abstract polynomials of degree less than or equal to n. We then recover, in the abstract setting, classical estimates of remainders in polynomial interpolation and quadrature formulas. Finally, we present an application to the error analysis of the trapezoidal time discretization scheme for parabolic evolution equations.
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.
Improving a recent result by the authors, a vector-valued version of Peano's Kernel Theorem is proposed, which gives an integral estimate for a class of linear operators L : Cn+1([a, b];X) --> X, with X general...
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Improving a recent result by the authors, a vector-valued version of Peano's Kernel Theorem is proposed, which gives an integral estimate for a class of linear operators L : Cn+1([a, b];X) --> X, with X general normed space, vanishing on all abstract polynomials of degree less than or equal to n. Continuity and derivatives are intended in the weak sense. When the space is complete, the usual integral representation is retrieved. We show that all usual remainder estimates for polynomial, piecewise polynomial, and spline polynomial interpolation, numerical differentiation and numerical quadrature, can be readily transferred in the vector-valued setting.
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...
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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),...
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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 consider spaces of continuous vector-valued functions on a locally compact Hausdorff space, endowed with classes of locally convex topologies, which include and generalize various known ones such as weighted space-...
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We consider spaces of continuous vector-valued functions on a locally compact Hausdorff space, endowed with classes of locally convex topologies, which include and generalize various known ones such as weighted space- or inductive limit-type topologies. The main result states that every continuous linear functional on such a function space can be expressed as an integral with respect to some canonical (dual space-valued) vector measure.
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...
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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...
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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.
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