Stancu-Kantorovich算子在Ba空间的逼近

252Vol.25No.2 20024ACTA MATHEMATICAE APPLICATAE SINICA Apr.,2002 Stancu-KantorovichBa∗(100875)(411201)Stancu-Kantorovich BaStancu-Kantorovich Ba1Ba(Laplace)1[1]B={L p1,L p2,···,L pm,···}Lebesgue f(x)nR n G a={a1,a2,···a m,···}f(x)∈∩m L pm,α>0I(f,α):=∞m=1a mαm f m pm<+∞,f(x)∈Ba,f Ba:=infα>0α:If,1α≤1Ba Ba Banach[1,2].B={L p,L p,···,L p,···},a={1,0,···,0,···},Ba Lebesgue L p.B={L1,L2,···,L m,···},Ba Orlicz L(φ)([1]).Ba L p Orlicz[1,3] Ba OrliczBa A b,20009202001725∗304252[4]1LebesgueB ,LebesgueA ={L q 1,L q 2,···,L q m ,···},1p m +1q m=1,m =1,2,···.b ={b 1,b 2,···,b m ,···}af (x )R nG 1GBaA b .f (x )[0,1]Stancu-KantorovichK n,s (f,x )=n k =0q n,k,s (x )(n +1)I kf (t )d t,x ∈[0,1],I k =k n +1,k +1n +1,0≤s <n2q n,k,s (x )=⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ n −s k x k (1−x )n −s −k +1,0≤k <s, n −s k x k (1−x )n −s −k +1+n −sk −s x k −s +1(1−x )n −k ,s ≤k ≤n −s, n −s k −sx k −s +1(1−x )n −k ,n −s <k ≤n.K n,s (f,x )s =0s =1K n,s (f,x )Bernstein-Kantorovich[5][4]BaBaStancu-Kantorovichf (x )∈Ba [0,1]t >0,ω2(f,t )Ba =sup 0≤h ≤tf (x +h )+f (x −h )−2f (x )Ba .α>0,C (r,q,···)1B ={L p 1,L p 2,···,L p m ,···}Lebesgue p m >1(m =1,2,···),a ={a 1,a 2,···,a m ,···}{a 1/m m },{a −1/mm }∈l ∞,inf mp m =p 0>1,f (x )∈Ba [0,1]nK n,s (f )−f Ba ≤C (r,q,p 0)ω2 f, (n +s 2−s −1)/(n +1)2Ba ,r =inf m{a 1/m m },q =sup m{a 1/mm }.2Ba A b ,{a 1/mm },{a −1/mm }∈l ∞,{b 1/mm },{b −1/m m }∈l ∞,r =inf m{a 1/m m },q =sup m{a 1/m m },r =inf m{b 1/m m },q =sup m{b 1/m m },2r ≤qq ,2r ≤qqK n,s Ba(i) K n,s (f )−f Ba =o 1n ⇔f (x )=const.;(ii) K n,s (f )−f Ba =O 1n⇒f (x )∈S B ;2Stancu-KantorovichBa305(iii)p 0=inf mp m >1,f (x )∈S BK n,s (f )−f Ba =O 1n,S B =g (x ):g (x )=const.+ x 01t (1−t ) t 0h (u )d u d t ,h (x )∈Ba [0,1] 1h (x )d x =0 .21K n,s (f )(i)K n,s Ba [0,1] K n,s ≤q 4r ;(ii)K n,s (1,x )=1;(iii)K n,s (t −x,x )=12(n +1)−xn +1;(iv)K n,s (t −x )2,x =13(n +1)2+n +s 2−s −1(n +1)2x (1−x );(v)K n,s |t −x |,x ≤K n,s (t −x )2,x 1/2.nk =0q n,k,s (x )=1,(ii),(iii),(iv)(v)Schwarz (i):K n,sp m >1(m =1,2,···),1p m+1q m=10≤s <n 2,K n,s (f ) p m=1nk =0q n,k,s (x )(n +1) I kf (u )d up m d x1/p m=10nk =0q 1/q m n,ks (x )q 1/p mn,k,s (x )(n+1)I kf (u )d up m d x1/p m≤ 10nk =0q n,k,s (x )p m /q m nk =0q n,k,s (x )(n +1)I kf (u )d up m d x1/p m = 10nk =0q n,k,s (x ) I k(n +1)f (u )d up m d x1/p m≤10n k =0q n,k,s (x )(n +1)I k|f (u )|p md u d x=n k =010q n,k,s (x )d x (n +1)I k|f (u )|p md u1/p m ≤ n +1n −s +1I |f (u )|p md u 1/p m≤2 f p m ≤2r f Ba . 10q n,k,s(x )d x ≤1n −s +1([7]) f p m ≤1r f Ba ([2]),K n,s (f ) Ba=infα>0:∞ m =1a m αm K n,s (f ) mp m ≤1 ≤infα>0:∞ m =1a m αm 2r f Ba m ≤1 ≤inf α>0:∞ m =12q αr f Ba m ≤1 .α=4qr f Ba ,∞ m =12qαr f Bam=1.K n,s (f ) Ba≤4qr f Ba ,k n,s ≤4q r .1306252[5]1f (x )∈Ba [0,1]θf (x )∈Ba [0,1], θf Ba ≤q 4r p 0p 0−1 f Ba ,θf (x )=sup0≤t ≤1,t =x1t −x t x |f (u )|d uf (x )Hardy-Littlewood f (x )∈Ba [0,1],f (x )[0,1]x /∈[0,1]f (x )=0.f h (x )=12h2h2−h 2h 2−h 2f (x +u +v )+f (x −u −v )d u d v,3[5]f (x )∈Ba [0,1](i)fh (x )∈Ba [0,1];(ii) f h (·)−f (·) Ba ≤qr ω2(f,h )Ba ;(iii) fh (·) Ba ≤2q hr ω2(f,h )Ba ;(iv) fh (·) Ba ≤4q h 2r ω2(f,h )Ba .4[7]f (x )∈C 2[0,1],lim n →∞n K n,s (f,x )−f (x ) = x (1−x )f (x ).5f (x )∈Ba [0,1],g (x )∈C 2[0,1],A n (f,g )=n1K n,s (f,x )−f (x )g (x )d x,C (g )>0,A n (f,g ) ≤C (g ) f Ba .[6]5.3C (g )>0,A n (f,g ) ≤C (g ) f L ≤C (g ) f Ba .6[2]2p 0=inf m{p m },p ∗=sup m{p m },L P ∗⊂Ba ⊂L p 0,BaL p ∗L p 0θ-72F (x )∈Ba [0,1],T n (F,x )=(n +1)K n,sx tg (u )−g (t )F (u )d u,x ,T n (F,x )Ba=OF Ba ,g (t )=ln(t )−ln(1−t ).Q (n,t,x )=nk =0q n,k,s (x )(n +1)χI k (t ),χI kI kg (u )−g (t )+=g (u )−g (t ),0≤t ≤u ;0,u <t ≤1,T n (f,·) 1=1T n (F,x ) d x =1(n +1)K n,sxtg (u )−g (t ) F (u )d u,xd x=1(n +1)nk =0q n,k,s (x )(n +1)I kxt g (u )−g (t ) F (u )d u d td x2Stancu-Kantorovich Ba307=10(n+1)1Q(n,t,x)xtg(u)−g(t)F(u)d u d td x≤ 1(n+1)xQ(n,t,x)xtg(u)−g(t)|F(u)|d u d t d x +1(n+1)1xQ(n,t,x)xtg(u)−g(t)F(u)d u d td x=(n+1)10 xuQ(n,t,x)g(u)−g(t)d t|F(u)|d u d x+10 1x1uQ(n,t,x)g(t)−g(u)d t|F(u)|d u d x≤(n+1) 11uuQ(n,t,x)g(u)−g(t)d t d x+u0 1uQ(n,t,x)g(u)−g(t)d t d x|F(u)|d u=(n+1)10 1uK n,s(g(u)−g(·))+,xd x+u0K n,s(g(·)−g(u))+,xd x|F(u)|d u≤(n+1)2n−s+1 1uK n,sg(·)−g(u),x−g(x)−g(u)d x|F(u)|d u≤(n+1)2n−s+1 1uK n,s(g,x)−g(x)d x|F(u)|d u≤(n+1)2n−s+1O1n1|F(u)|d u=OF 1.K n,s(g)−g 1=O1n([7,3]),1u K n,s(g(u)−g(·))+,xd x≤ uK n,sg(x)−g(u),xd x+n+1n−s+1ug(u)−g(x)d x,10K n,s(g(u)−g(·))+,xd x≤n+1n−s+1ug(u)−g(x)+d x=n+1n−s+1ug(u)−g(x)d x.T n(F,x)=(n+1)K n,sxtg(u)−g(t)F(u)d u,x30825=(n+1)nk=0q n,k,s(x)I kxtg(u)−g(t)F(u)d u d t=(n+1)1Q(n,t,x)xtg(u)−g(t)F(u)d u d t≤ F ∞(n+1) 1Q(n,t,x)xtg(u)−g(t)d u d t= F ∞(n+1)xln x−K n,s(ln(·),x)+(1−x)ln(1−x)−K n,s(ln(1−·),x)= F ∞(n+1)O 1n=OF ∞.M.Riesz-Thorin6F(x)∈Ba[0,1]T n(F,x)Ba=OF Ba.731(1)f (x)∈Ba[0,1],ξ∈(x,t)f(t)−f(x)=(t−x)f (ξ).1,max0≤x≤1|1−2x|=1,max0≤x≤1|x(1−x)|=14,nK n,s(f,x)−f(x)=(n+1)nk=0q n,k,s(x)I kf(t)−f(x)d t=(n+1)nk=0q n,k,s(x)I k(t−x)f (ξ)d t=(n+1)nk=0q n,k,s(x)I k(t−x)f (x)+(t−x)f (ξ)−f (x)d t=(n+1)nk=0q n,k,s(x)I k(t−x)f (x)+(t−x)2t−xξxf (u)d ud t≤(n+1)nk=0q n,k,s(x)I k(t−x)d t·|f (x)|+(n+1)nk=0q n,k,s(x)I k(t−x)2d t·θf (x)=|K n,s(t−x,x)|·|f (x)|+K n,s(t−x)2,xθf (x)=|1−2x|2(n+1)|f (x)|+13(n+1)2+n+s2−s−1(n+1)2x(1−x)·|θf (x)|≤12(n+1)|f (x)|+n+s2−s−1(n+1)2θf (x).2K n,s(f)−f pm ≤12(n+1)f pm+n+s2−s−1(n+1)2θf pm2Stancu-KantorovichBa309≤12r (n +1) fBa +n +s 2−s −1r (n +1)2θf Ba≤12r (n +1) f Ba +q (n +s 2−s −1)2r (n +1)2p 0p 0−1f Ba . K n,s (f )−f Ba ≤infα>0:∞ m =1a m αm K n,s (f )−f mp m ≤1 ≤inf α>0:∞ m =1a m αm 12r (n +1) f Ba +q (n +s 2−s −1)2r 2(n +1)2p 0p 0−1 fBa m ≤1 ≤infα>0:∞ m =11αm q 2r (n +1) f Ba +q 2(n +s 2−s −1)2r 2(n +1)2p 0p 0−1 fBa m ≤1 ≤q r (n +1) fBa +q 2(n +s 2−s −1)r 2(n +1)2p 0p 0−1 f Ba .(2)f (x )∈Ba [0,1],(1)13K n,s (f )−f Ba = K n,s (f −f h )+K n,s (f h )−f h +f h −f Ba≤ K n,s (f −f h ) Ba + K n,s (f h )−f h Ba + f h −f Ba ≤ 1+q 4r q r ω2(f,h )Ba +q r (n +1)2qhr ω2(f,h )Ba +4q 3(n +s 2−s −1)r 3h (n +1)2p 0p 0−1ω2(f,h )Ba .h = n +s 2−s −1(n +1)2,C (r,q,p 0)= 1+q 4r q r +2q 2r 2+4q 3r 3p 0p 0−1,K n,s (f )−f Ba=C (r,q,p 0)ω2 f, (n +s 2−s −1)/(n +1)2Ba .1245f (x )∈C 2[0,1]∩Ba [0,1]g (x )∈C 2[0,1],lim n →∞A n (f,g )=lim n →∞n1K n,s (f,x )−f (x )g (x )d x=10lim n →∞n K n,s (f,x )−f (x )g (x )d x =10 x (1−x )f (x )g (x )d x=10x (1−x )g (x )f (x )d x.C 2[0,1]∩Ba [0,1]Ba [0,1]lim n →∞A n (f,g )=1x (1−x )g (x )f (x )d x,f (x )∈Ba [0,1](i) K n,s (f )−f Ba =o 1n, 1(x (1−x )g (x )) f (x )d x =0.f (x )=0(a.e.),f (x )=const.31025f(x)=const. K n,s(f)−f Ba=o1 n.(ii) K n,s(f)−f Ba=O1n,[4,1][4,7]h(x)∈Ba[0,1]1 0x(1−x)g (x)f(x)d x=1g(x)h(x)d x.f(x)=const.+x1t(1−t)th(u)d u d t,1h(u)d u=0.f(x)∈S B.(iii)f(x)∈S B,f (x)=1x(1−x) xh(u)d u,x(1−x)f (x)=xh(u)d u.1 0h(x)d x=0xh(u)d u=x1h(u)d u,h(x)∈Ba[0,1]2(1−x)f (x),xf (x)∈Ba[0,1],(1−2x)f (x)+x(1−x)f (x)=h(x)x(1−x)f (x)∈Ba[0,1].g(t)=ln t−ln(1−t),f(t)−f(x)=−xt f (u)d u=x(1−x)f (x)g(t)−g(x)+xtg(u)−g(t)du(1−u)f (u).K n,s(f,x)−f(x)=x(1−x)f (x)K n,s(g,x)−g(x)+K n,sxtg(u)−g(t)du(1−u)f (u),x.[7,3] K n,s(g)−g 1=O(1n),x(1−x)K n,s(g,x)−g(x)1=O1.[8,1]x(1−x)K n,s(g,x)−g(x)∞=O1n.Riesz-Thorin6x(1−x)K n,s(g,x)−g(x)Ba=O1n.T n(F,x)=(n+1)K n,sxtg(u)−g(t)F(u)d u,x.2Stancu-Kantorovich Ba311F(x)∈Ba[0,1],7 T n(F,x) Ba=OF Ba.F(x)=x(1−x)f (x),(n+1)K n,s xtg(u)−g(t)u(1−u)f (u)d u,xBa=Ox(1−x)f (x)Ba+(1−2x)f (x)Ba,K n,s xtg(u)−g(t)u(1−u)f (u)d u,xBa=O1n.K n,s(f)−f Ba=O1n.21Ding Xiaqi,Luo Peizhu.Ba Spaces and some Estimates of Laplace Operator.J.Sys.Sci.&Math.Sci.,1981,1(1):9–332Chen Guangrong,Meng Boqin.Interpolation of Ba Spaces.Math.Acta Sci.,1988,8(1):1–10 3Ba,1989,9(4):407–413(Zhuang Yadong,Yu Xintai.Some Properties of Ba Spaces.Math.Acta Sci.,1989,9(4):407–413) 4Ba Kantorovich,1992,12(2):146–154 (Sheng Baohuai.The Saturation of Kantorovich Operators in Ba Spaces.J.of Math.(PRC),1992, 12(2):146–154)5Ba Kantorovich,1996,1:7–11 (Wu Garidi,Chen Guangrong.Approximation by Kantorovich Operators in Ba Spaces.J.of Inner Mongolia Normal University,1996,1:7–11)6Ditzian Z,May C P.L p Saturation and Inverse Theorems for Modified Bernstein Polynomials.Indiana Univ.Math.J.,1976,25:733–7517Stancu-Kantorovich L p,1988,8(3):257–262 (Zhao Jinghui.The L p Saturation of Stancu-Kantorovich Operators.J.of Math.(PRC),1988,8(3): 257–262)8Riemenschneider S D.The L p-saturation of the Bernstein-Kantorovich Polynomials.J.of Approx.Theory,1978,23:158–162APPROXIMATION BY STANCU-KANTOROVICHOPERATORS IN Ba SPACESWU Huoxiong(Department of Mathematics,Beijing Normal University,Beijing100875)ZHU Lizhi(Department of Mathematics,Xiangtan Normal College,Xiangtan411201)Abstract The approximation problem by Stancu-Kantorovich operators in Ba spaces is studied.An estimation of degree of approximation and the saturation theorems are given. Key words Stancu-Kantorovich operators,Ba spaces,approximation,saturation。