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1、应用数学MATHEMATICA APPLICATA2023,36(4):903-914A Jackson Inequality for KernelFunction Approximation in LearningTheoryTIAN Mingdang(田明党),SHENG Baohuai(盛宝怀)(Department of Economic Statistics,Zhejiang Yuexiu University,Shaoxing 312000,China)Abstract:We recognize kernel function spaces from the view of dif
2、ferential operatorsand discuss the kernel function approximation problem with the classical Fourier transform.We define a modulus of smoothness with the Fourier multiplier operators associated withthe semigroup of operators and show that it is equivalent to a K-functional defined witha given kernel
3、based differential operator,with which we provide a classical Jackson-typeinequality to describe the decay of best kernel function approximation.We show that ifthe differential operator is the Riesz potential operator or the Bessel potential operator,then the decay can be bounded with the modulus of
4、 smoothness defined by a convolutionaloperator.In particular,we give an upper bound estimate for the best approximation by areproducing kernel Hilbert space.Key words:Jackson-type inequality;K-functional;Modulus of smoothness;Reproduc-ing kernel Hilbert space;Riesz potential operator;Poisson kernel;
5、Learning theoryCLC Number:O174.41AMS(2010)Subject Classification:41A25Document code:AArticle ID:1001-9847(2023)04-0903-121.IntroductionKernel function approximation problems belong to the mathematical foundation of statis-tical learning and artificial intelligence12,among which there is the followin
6、g approximationproblem34:Let(B,)be a Banach space and(H,H)be a dense subspace with embeddingrelationb k bH,b H,(1.1)where k 0 is a given constant independent of b.Given a B,what is the convergence rateof the functionIH,H(a,)=infbH,bH(a b),0,(1.2)as +.Received date:2022-09-10Foundation item:Supported
7、 partially by the NSF(61877039),the NSFC/RGC Joint ResearchScheme of China(12061160462 and NCityU102/20)and the NSF of Zhejiang Province(LY19F020013)Biography:TIAN Mingdang,female,Han,Shandong,lecturer,major in statistical learning theory.904MATHEMATICA APPLICATA2023A typical example is when B is th
8、e square integrable function space5.Let X be acomplete metric space and be a Borel measure on X.Denoted by L2(X)the Hilbert spaceconsisting of(real)square integrable functions with the inner productf,gL2(X)=Xf(x)g(x)d(x),f,g L2(X).Suppose that K:X X R is continuous,symmetric and strictly positive de
9、finite,i.e.(K(xi,xj)mi,j=1is a positive definite matrix for any given finite sets x1,x2,xm X.Assume that K L2(X X),i.e.,XX|K(x,t)|2d(x)d(t)+.Then the integraloperator LK:L2(X)L2(X)defined byLK(f,x)=XK(x,t)f(t)d(t),x X,(1.3)is positive definite associated with the kernel K(x,y),and its range lies in
10、C(X).Take L12Ktobe the linear operator on L2(X)satisfying L12K L12K=LKand L12Kthe inverse of L12K.Theoperator L12Khas many nice properties the same as that of derivatives,therefore it can beregarded as a differential operator6.For a given positive integer r we define H(r)K=Lr2K(L2(X)and equip which
11、with normfH(r)K=Lr2KfL2(X),f H(r)K,i.e.,Lr2KfH(r)K=fL2(X),f L2(X).Since f=Lr2K(Lr2K(f),we havefL2(X)=Lr2K(Lr2K(f)L2(X)Lr2KL2(X)L2(X)Lr2K(f)L2(X)L12KrL2(X)L2(X)fH(r)K.The boundedness of L12KL2(X)L2(X)is a critical quantity for describing the operator Lr2K,and we provide an assumption about it.Assumpt
12、ion 1L12Kis a bounded operator from L2(X)to L2(X)andL12KL2(X)L2(X)+.(1.4)If(1.4)holds,then we have the following embedding inequalityfL2(X)crfH(r)K,f H(r)K,(1.5)where c=L12KL2(X)L2(X)0.(1.7)We give two examples.Example 1The Riesz potential operator7I(f)(x):=I(f,x)=1()Rdx ydf(y)dy,()=d22(2)(d2),x Rd(
13、1.8)No.4TIAN Mingdang,et al.:A Jackson Inequality for Kernel Function Approximation905is an integral operator associated with kernel the K(x,y)=x yd.Example 2The Bessel kernel7G(x)=1(4)21(2)+0ex2e4d+2d,x Rd,(1.9)satisfies for 0,G(x)L1(Rd).The Bessel potential operator is defined asB(f)(x):=(G f)(x)=
14、RdG(x y)f(y)dy,x Rd.(1.10)For a given 0,B(f)is a kernel based integral operator with K(x,y)=G(x y).It is known that,to describe the decay of best approximation by algebraic polynomialsof order nEn(f)L2(1,1)=infpPnf pL2(1,1),f L2(1,1),(1.11)where Pndenotes the set of all the algebraic polynomials of
15、order n,one establishes theclassical Jackson-type inequality8En(f)L2(1,1)Cr(f,1n)L2(1,1),f L2(1,1),(1.12)where r(f,t)L2(1,1)is the r-th Totik-Ditzian modulus of smoothness.To describe the decay of best approximation by the entire function of exponential type,one needs the Jackson inequality9E(f)L2(R
16、)Cr(f,1)L2(R),f L2(R),(1.13)where r(f,t)L2(R)is the r-th modulus of smoothness defined byr(f,t)L2(R)=sup|h|trhf()L2(R),rhf(x)=rj=0(1)jCjrf(x+jh),Cjr=r!j!(r j)!.The aim of the present paper is to establish a Jackson-type inequality to describe thedecay of I(a,R)when B=L2(Rd),and H(r)K=Lr2K(L2(Rd)with
17、 K(r)being some positivedefinite kernels defined on Rd.We consider now a more general problemIH(r)K,H(r)K(f,)L2(R)=infgH(r)K,Lrv2KgL2(Rd)(f gL2(Rd),0,(1.14)as +,where are no-negative integers and 0 0,(1.15)where the K-functional Dr(f,t)L2(Rd)is defined asDr(f,t)L2(Rd)=infgH2,(f gL2(Rd)+trIrgL2(Rd),f
18、 L2(Rd),t 0,(1.16)where H2,r=g L2(Rd):Irg L2(Rd).The modulus of smoothness r(f,t)L2(Rd)isdefined byr(f,t)L2(Rd)=sup|h|trhf()L2(Rd).906MATHEMATICA APPLICATA2023In the present paper we shall provide a modified equivalent relation similar to(1.15)withthe help of Fourier multipliers and with which show
19、a Jackson inequality to describe the bestapproximation(1.7).Replacing the Riesz potential operators Irin(1.16)with the generaloperators Lr2K,we have the following r-th K-functionalDrLK(f,t)L2(Rd)=infgLr2K(L2(Rd)(f gL2(Rd)+trLr2KgL2(Rd)=infgH(r)K(f gL2(Rd)+trgH(r)K),f L2(Rd),t 0,(1.17)where H(r)K=Lr2
20、K(L2(Rd)and gH(r)K=Lr2K(g)L2(Rd).When r=1,we have the K-functional defined in learning theory6,1113DHK(f,t)L2(Rd)=infgHK(f gL2(Rd)+t gHK),f L2(Rd),t 0,(1.18)where HK=H(1)K=L12K(L2(Rd)is the reproducing kernel Hilbert space associated withreproducing kernel K(x,y).We shall show that the r-th K-functi
21、onal DrLK(f,t)L2(Rd)is e-quivalent to an r-th modulus of smoothness and,with which provide a Jackson-type inequalityfor the decay of(1.14).The manuscript is organized as follows.In Section 2 we shall redefine the kernel func-tion space H(r)Kfor some kernel functions expressed with the Fourier transf
22、orms and define amodulus of smoothness with the semi-group of operators associated with the Fourier multipli-ers,which is equivalent to the K-functional DrLK(f,t)L2(Rd)and give the concrete equivalentrelation for the Riesz potential operator,the Bessel potential operator and a general repro-ducing k
23、ernel Hilbert space defined by the Fourier transforms.In Section 3,we provide aJackson-type inequality for the best kernel function approximation and show that,in thecases of Riesz operator and Bessel operator,the decay may be bounded by a convolutionaloperator.In Section 4 we shall restate a genera
24、l equivalent relation between a K-functionaland a modulus of smoothness,which will be used to prove the main results in Section 2.Throughout the paper,we shall write A=O(B)if there exists a constant C 0 suchthat A CB.We write A B if A=O(B)and B=O(A).Also we denote by N the set ofnon-negative positiv
25、e integers.2.K-Functionals and Moduli of SmoothnessFor x Rdand =(1,d)Zd+we define x=x11xdd,|=1+dand(x)=|x11xdd.The Schwartz space S(Rd)consists of all indefinitely differentiable functions f on RdsuchthatsupxRd?x(x)f(x)?0 and there is an even function L1(Rd)such that 12()=().Under this assumption,th
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