学习理论中核函数逼近的Jackson型不等式  

A Jackson Inequality for Kernel Function Approximation in Learning Theory

作　　者：田明党 盛宝怀 TIAN Mingdang;SHENG Baohuai(Department of Economic Statistics,Zhejiang Yuexiu University,Shaoxing 312000,China)

机构地区：[1]浙江越秀外国语学院经济统计系,浙江绍兴312000

出　　处：《应用数学》2023年第4期903-914,共12页Mathematica Applicata

基　　金：Supported partially by the NSF(61877039);the NSFC/RGC Joint Research Scheme of China(12061160462 and N_CityU102/20);the NSF of Zhejiang Province(LY19F020013)。

摘　　要：从微分算子角度理解核函数空间,借助经典Fourier变换研究核函数逼近问题.应用Fourier乘子算子和算子半群定义了一种光滑模,证明其与一种基于微分算子的K-泛函的等价性,由此给出了刻画核函数逼近收敛性的Jackson不等式.进一步证明,如果微分算子为Riesz势算子或Bessel势算子,逼近的收敛性可以转化为卷积算子逼近.特别地,给出了再生核Hilbert空间逼近的一种上界估计.We recognize kernel function spaces from the view of differential operators and discuss the kernel function approximation problem with the classical Fourier transform.We define a modulus of smoothness with the Fourier multiplier operators associated with the semigroup of operators and show that it is equivalent to a K-functional defined with a given kernel based differential operator,with which we provide a classical Jackson-type inequality to describe the decay of best kernel function approximation.We show that if the differential operator is the Riesz potential operator or the Bessel potential operator,then the decay can be bounded with the modulus of smoothness defined by a convolutional operator.In particular,we give an upper bound estimate for the best approximation by a reproducing kernel Hilbert space.

关 键 词：Jackson不等式 K-泛函 光滑模 再生核HILBERT空间 Riesz势算子 POISSON核 学习理论 

分 类 号：O174.41[理学—数学]