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KorovkinWeierstrassKorovkinf(x)[a,b]Pn(x)[a,b]f(x)f(x)f(x)[a,b]0P(x)P(x)f(x)x[a,b]Korovkin1953(Weierstrass)f(x)[a,b]0P(x)P(x)f(x)x[a,b][a,b][0,1]X[0,1]YBn:X→Yf(t)B6n(f,x)=∑=−−nkknkknxxnkf0)1(C)(Bn(f,x)fXBnxnBernsteinBn(1)Bnf,gX,RBn(f+g,x)=Bn(f,x)+Bn(g,x)(2)Bnf,gXf(t)g(t)t[a,b]Bn(f,x)Bn(g,x)x[a,b](3)Bn(1,x)==[x+(1x)]∑=−−nkknkknxx0)1(Cn=1Bn(t,x)=∑=−−nkknkknxxnk0)1(C=x∑=−−−−−nkknkknxx1111)1(C=x[x+(1x)]n=xBn(t2,x)=∑=−−nkknkknxxnk022)1(C=∑=−−−−nkknkknxxnk111)1(C=∑=−−−−−nkknkknxxnk211)1(C1+∑=−−−−nkknkknxxn111)1(C1=∑=−−−−−−nkknkknxxxnn22222)1(C1+∑=−−−−−nkknkknxxnx1111)1(C=21xnn−+nx=+2xnxx2−(ts)2BnsBn((ts)2,x)=Bn(t2,x)2sBn(t,x)+s2Bn(1,x)=x2+nxx2−2sx+s2=nxx2−+(xs)2f[0,1]M0t[0,1]f(t)MCantorf[0,1]00t,s[0,1]tsf(t)f(s)2ε;tsf(t)f(s)2M22δM(ts)2t,s[0,1],2ε22δM(ts)2f(t)f(s)2ε22δM(ts)2(t)Bn(x)f(s)Bn(f(s),x)=f(s)(1)(2)(3)x,s[0,1]2ε22δM[nxx2−+(xs)2Bn(f,x)f(s)2ε22δM[nxx2−+(xs)2s=xx(1x)41,∑=−−−⎟⎠⎞⎜⎝⎛nkknkknxfxxnkf0)()1(C2ε22δnMN=[εδ2M]nN,∑=−−−⎟⎠⎞⎜⎝⎛nkknkknxfxxnkf0)()1(Cx[0,1]10.5.1f[a,b]BernsteinBn(f,x)[a,b]f1f(x)[a,b]f(x)=,x[a,b]∑∞=−00)(nnnxxaSn(x)=∑=−nkkkxxa00)(f(x)[a,b]Sn(x)nnSn(x)n1Sn1(x)an(xx0)nSn1(x)Weierstrass[a,b]f(x)2Bernstein