Approximating properties of metric and generalized metric projections in uniformly convex and uniformly smooth Banach spaces

IF 0.9 3区数学 Q2 MATHEMATICS Journal of Approximation Theory Pub Date : 2023-09-27 DOI:10.1016/j.jat.2023.105973

Akhtar A. Khan , Jinlu Li

引用次数: 5

Abstract

This note conducts a comparative study of some approximating properties of the metric projection, generalized projection, and generalized metric projection in uniformly convex and uniformly smooth Banach spaces. We prove that the inverse images of the metric projections are closed and convex cones, but they are not necessarily convex. In contrast, inverse images of the generalized projection are closed and convex cones. Furthermore, the inverse images of the generalized metric projection are neither a convex set nor a cone. We also prove that the distance from a point to its projection on a convex set is a weakly lower semicontinuous function for all three notions of projections. We provide illustrating examples to highlight the different behavior of the three projections in Banach spaces.

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一致凸一致光滑Banach空间中度量和广义度量投影的逼近性质

本文比较研究了一致凸和一致光滑Banach空间中度量投影、广义投影和广义度量投影的一些逼近性质。我们证明了度量投影的逆像是闭凸锥，但它们不一定是凸的。相反，广义投影的逆像是闭的凸锥。此外，广义度量投影的逆像既不是凸集也不是锥。我们还证明了从一个点到它在凸集上的投影的距离对于所有三个投影概念都是弱下半连续函数。我们提供了示例来强调Banach空间中三个投影的不同行为。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Approximation Theory 数学-数学

CiteScore

1.90

自引率

11.10%

发文量

审稿时长

6-12 weeks

期刊介绍： The Journal of Approximation Theory is devoted to advances in pure and applied approximation theory and related areas. These areas include, among others: • Classical approximation • Abstract approximation • Constructive approximation • Degree of approximation • Fourier expansions • Interpolation of operators • General orthogonal systems • Interpolation and quadratures • Multivariate approximation • Orthogonal polynomials • Padé approximation • Rational approximation • Spline functions of one and several variables • Approximation by radial basis functions in Euclidean spaces, on spheres, and on more general manifolds • Special functions with strong connections to classical harmonic analysis, orthogonal polynomial, and approximation theory (as opposed to combinatorics, number theory, representation theory, generating functions, formal theory, and so forth) • Approximation theoretic aspects of real or complex function theory, function theory, difference or differential equations, function spaces, or harmonic analysis • Wavelet Theory and its applications in signal and image processing, and in differential equations with special emphasis on connections between wavelet theory and elements of approximation theory (such as approximation orders, Besov and Sobolev spaces, and so forth) • Gabor (Weyl-Heisenberg) expansions and sampling theory.

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