A Further Research on the Convergence of Wu-Schaback’s Multi-quadric Quasi-Interpolation

Quasi-interpolation methods have been used widely in data analysis, and have great values not only in theory but also in many application areas such as medicine, geology, economy and computer science. Multiquadric functions were first proposed by Hardy [1] in 1968, and Franke [2] showed they performed well in many calculations including the numerical experiments. Powell [3], Beatson and Powell [4], and Beatson and Dyn [5] successively proposed a number of quasiinterpolation schemes and discussed the convergence of the schemes. In 1994, Wu and Schaback [6] proposed a useful quasi-interpolation operator D L f and discussed the convergence and shape preserving properties of this operator. In their convergence theorem (theorem A in our paper), they claimed interpolated functions 2 ( ) f x C ∈ . Based on these papers, Zhang and Wu [7], and Ma and Wu [8] did further researches. In this paper, we discuss the convergence of operator D L f for a wider range of approximated functions (namely functions with lower smoothness). To prove the convergence, we use two theorems showed by Beatson and Powell [4], and our method differs from that in [6].


Introduction
Quasi-interpolation methods have been used widely in data analysis, and have great values not only in theory but also in many application areas such as medicine, geology, economy and computer science. Multiquadric functions were first proposed by Hardy [1] in 1968, and Franke [2] showed they performed well in many calculations including the numerical experiments. Powell [3], Beatson and Powell [4], and Beatson and Dyn [5] successively proposed a number of quasiinterpolation schemes and discussed the convergence of the schemes. In 1994, Wu and Schaback [6] proposed a useful quasi-interpolation operator D L f and discussed the convergence and shape preserving properties of this operator. In their convergence theorem (theorem A in our paper), they claimed interpolated functions 2 ( ) f x C ∈ . Based on these papers, Zhang and Wu [7], and Ma and Wu [8] did further researches. In this paper, we discuss the convergence of operator D L f for a wider range of approximated functions (namely functions with lower smoothness). To prove the convergence, we use two theorems showed by Beatson and Powell [4], and our method differs from that in [6].

Preparation
We assume that there are finite scattered points 0 { } N j j x = in the bounded interval [a,b] as follows: and the maximum spacing is defined as , we define its norm as and its modulus of continuity as The basis functions used in this paper are where c>0 is a positive shape parameter In 1994, Wu and Schaback proposed the quasi-interpolation operator L D : They got the error estimate of this operator as follows: the quasi-interpolant D L f satisfies an error estimate of type where positive constants 1 2 3 , , K K K are independent of h and c.
In 1992, Beatson and Powell [4] proposed the quasi-interpolation operator L B : They proved the following result: [4] the quasi-interpolation operator L C was defined as follows: They got the following theorem: Theorem C: If f has a Lipschitz continuous first-order derivative, then the maximum error of the quasi-interpolant C L f satisfies the bound 2 2

Main Result
It should be noticed that in Theorem A, Wu and Schaback demanded the approximated function 2 [ , ] f C a b ∈ . In this paper, we weaken this condition step by step. Using Theorem B and Theorem C proposed by Beatson and Powell, we get three theorems about convergence estimate for the approximated functions with lower smoothness.
Theorem 1: If f has a Lipschitz continuous first-order derivative, then we can draw the conclusion: Proof: We notice that quasi-interpolant D L f and B L f have the following relationship: In [4], Beatson and Powell have showed the relationship between B L f and C L f : where , we can easily get the two inequalities: Using (1), (2),(3), we can get Further, due to Theorem C, we can get 2 2 || || || || || ||  (3), it is obvious that