Approximation Of Smooth Function Based On Discrete Information

This paper mainly investigate approximation of smooth function based on discrete information and divides to three chapters.The first chapter be foreword.The second chapter be part of approximation by interpolation.It discuss the properties of the relative derivative and investigate Hermite interpolation for continuous piecewise smooth functions.A new family of Hermite type interpolatory functions with explicit approximation error bounds is obtained.The third chapter be part of approximation by operators.Let x∈[0,∞),p_n,k(x)= (-x)^k/k!φ_n^(k)(x) and(ⅰ) c＞0,φ_n(x)=(1+cx)^-n/c;(ⅱ) c=0,φ_n(x)=e^-nx,J_n,k(x)= (?)p_n,j(x);Q_n,k^a(x)=J_n,k^a(x)-J_n,k+1^a(x).In this chapter,we obtain the rate of convergence of the operator V_n,a(f,x)=(?)f(k/n)Q_n,k^a(x) and D_n,a(f,x)=(?)Q_n,k^a(?)b_n,k(t)f(t) dt with bounded variation.Where,D_n,a(f,x) be simultaneous approximation.