Some New Ideas. Operator Approximation In The Constructive Proof

The approximation theory of functions began in the end of 19th century, and was prosperous in the 20th century. Ever since then, it focused its target on using the simple and computable functions to approximate general functions, considered the degree of convergence and how to characterize the inherent peculiarity of functions. From then on , constructive property gradually becomes a tendency in approximation theory.It is very important in mathematics to create constructive ideas and some techniques deriving from them. In approximation theory of functions, for example, the elaborate construction of many operators generalizes good convergent rate as they approximate to some class of functions; the different construction of interpolation nodes makes distinction in approximation, etc..In this paper, we use some new ideas in the constructive proofs of operator approximation, and get the convergence or estimation of convergent order of some linear or nonlinear operators. For example, when we construct the interpolation nodes of rational functions, we find the relationship between the density of nodes to the singular point and the approximation order; we use interpolation nodes as the zeros of orthogonal polynomials corresponding to the exponential weights, and extend the range of convergence of operator to the whole real line, etc..Some notations will be introduced as follows: Let CI denote the real continuous function space defined on a finite or infinite interval I.For F ∈ CI, define||f||CI= max |f(x)|,andAlso let , denote the spaces of the p power integral functions on the interval [a, b], , for , defineandFor the interval [0, 1], for simplicity, we denote the above byUsually, a (7 indicates an absolute positive constants, (7, a positive constants depending only on s. They may represent different values at differernt occurences,even in the same line. And A ~ B means that there exists an absolute constant C > 0 such thatThis paper is divided into five chapters. In Chapter 1, we consider the approximation order of rational interpolation to |x|; in Chapter 2, we consider the special case of rational approximation- approximation by reciprocal of polynomials with positive coefficients in If spaces; and in Chapter 3, we consider the convergence of modified Bernstein operators; in Chapter 4, we consider the convergence of Griinwald operators interpolate the nodes as zeros of Hermite polynomials on the whole real line, and in Chapter 5, we consider the convergence of Hermite interpolation operators on the whole real line based on the above nodes.Chapter 1 The convergent order of rational interpolation to |x|It is well known that the best uniform approximation to \x\ by polynomials of degree n can achieve at the rate O(n-1), and this rate cannot be improved further(cf. [2]). However, in 1964, Newman [4] found that rational approximation to \x\ ( which interpolates \x\ at the nodes) is of much more benefit, namely, \x\ can be approximated uniformly by rational functions at an exponential rate.Later, many scholars like Werner [32],Bruteman,Passow [3, 4] considered Newman type interpolation at different nodes. The argument of these sets of nodes and the corresponding conclusions lead us to ask: why the optimal nodes in polynomial interpolation behave not so good in rational interpolation case? Why the " adjusted " nodes can get better order? What kind of nodes can be suitable for rational interpolation purpose?The present chapter constructs a set of nodes which can generate a rational interpolating function to approximate \x\ at the rate of O(1/(nk logn)) for any given positive integer k. Although the order concerned is worse than that in Newman's conclusion, it is more important that this construction reveals the fact that if the distribution of a set of nodes has higher density to zero (that is the singular point of the function |x|!), then the rational interpolation approximation behaves better.Following are the main results of this section,...