文献类型：期刊文献

英文题名：On the K-functional in learning theory

作者：Sheng, Bao-Huai[1];Wang, Jian-Li[1]

机构：[1]Shaoxing Univ, Dept Appl Stat, Shaoxing 312000, Zhejiang, Peoples R China

年份：2020

卷号：18

期号：3

起止页码：423

外文期刊名：ANALYSIS AND APPLICATIONS

收录：SCI-EXPANDED(收录号：WOS:000529067800003)、、WOS

基金：This work is supported by NSF (Project No. 61877039) of P. R. China and the NSF (Project No. LY19F020013) of Zhejiang Province. The authors are deeply grateful to the anonymous reviewers. Their review reports point out some grammatical errors, many redundant statements, some technical errors and notational incompleteness for the first manuscript, which help the authors make a careful revision and increase the readability of this paper.

语种：英文

外文关键词：Learning theory; K-functional; modulus of smoothness; spherical harmonics; Fourier series; convergence rate; the unit sphere; the unit ball

外文摘要：K-functionals are used in learning theory literature to study approximation errors in kernel-based regularization schemes. In this paper, we study the approximation error and K-functionals in L-p spaces with p >= 1. To this end, we give a new viewpoint for a reproducing kernel Hilbert space (RKHS) from a fractional derivative and treat powers of the induced integral operator as fractional derivatives of various orders. Then a generalized translation operator is defined by Fourier multipliers, with which a generalized modulus of smoothness is defined. Some general strong equivalent relations between the moduli of smoothness and the K-functionals are established. As applications, some strong equivalent relations between these two families of quantities on the unit sphere and the unit ball are provided explicitly.