摘要
Suppose that we want to approximate fC[0,1]by polynomials in P_n,using only its values on X_n={i/n,0≤i≤n}.This can be done by the Lagrange interpolant L_n f or the classical Bernstein polynomial B_n f.But,when n tends to infinity,L_n f does not converge to f in general and the convergence of B_n f to fis very slow.We define a family of operators B^(k)_n, n≥k,which are intermediate ones between B(0)_n=B^(1)_n=B_n and B^(n)_n=L_n,and we study some of their properties.In particular,we prove a Voronovskaja-type theorem which asserts that B^(k)_n f-f=0(n^(-[(k+2)/2))for f sufficiently regular. Moreover,B(k)_n f uses only values of B_n f and its derivaties and can be computed by De Casteljau or subdivision algorithms.
Suppose that we want to approximate fC[0,1]by polynomials in P_n,using only its values on X_n={i/n,0≤i≤n}.This can be done by the Lagrange interpolant L_n f or the classical Bernstein polynomial B_n f.But,when n tends to infinity,L_n f does not converge to f in general and the convergence of B_n f to fis very slow.We define a family of operators B^(k)_n, n≥k,which are intermediate ones between B(0)_n=B^(1)_n=B_n and B^(n)_n=L_n,and we study some of their properties.In particular,we prove a Voronovskaja-type theorem which asserts that B^(k)_n f-f=0(n^(-[(k+2)/2))for f sufficiently regular. Moreover,B(k)_n f uses only values of B_n f and its derivaties and can be computed by De Casteljau or subdivision algorithms.
