用单个函数和神经网络的移位和扩张的总和逼近

IF 0.9 3区数学 Q2 MATHEMATICS Journal of Approximation Theory Pub Date : 2023-07-01 DOI:10.1016/j.jat.2023.105915

K. Shklyaev

引用次数: 0

摘要

我们在函数f上找到了充分的条件，以确保形式为f（αx-θ）的函数的和，其中α∈a⊂R和θ∈θ\8834R，在实线上的实空间C0和Lp或其紧子集中是稠密的。也就是说，我们考虑所有系数都为1的线性组合。作为推论，我们推导出紧致子集上一致收敛拓扑中C（Rd）中函数f（w∙x-θ），w∈w⊂Rd，θ∈θ\8834R和的密度的结果。

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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Approximation by sums of shifts and dilations of a single function and neural networks

We find sufficient conditions on a function $f$ to ensure that sums of functions of the form $f (α x - θ)$ , where $α \in A \subset R$ and $θ \in Θ \subset R$ , are dense in the real spaces $C_{0}$ and $L_{p}$ on the real line or its compact subsets. That is, we consider linear combinations in which all coefficients are 1. As a corollary we deduce results on density of sums of functions $f (w \cdot x - θ)$ , $w \in W \subset R^{d}$ , $θ \in Θ \subset R$ in $C (R^{d})$ in the topology of uniform convergence on compact subsets.

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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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