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A Further Research on the Convergence of Wu-Schabacks Multi-quadric Quasi-Interpolation | OMICS International
ISSN: 2168-9679
Journal of Applied & Computational Mathematics
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A Further Research on the Convergence of Wu-Schabacks Multi-quadric Quasi-Interpolation

Yang Zhang, Xue-Zhang Liang, Qiang Li*

School of Mathematics, Jilin University, Changchun, 130012, P.R. China

*Corresponding Author:
Qiang Li
School of Mathematics, Jilin University
Changchun, 130012, PR China
E-mail: [email protected]

Received August 25, 2013; Accepted September 23, 2013; Published September 26, 2013

Citation: Zhang Y, Zhang Y, Liang XZ, Li Q (2013) A Further Research on the Convergence of Wu-Schaback’s Multi-quadric Quasi-Interpolation J Appl Computat Math 2: 138. doi: 10.4172/2168-9679.1000138

Copyright: © 2013 Zhang Y, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

The paper discusses the error estimate of Wu-Schaback's quasi-interpolant for a wider class of approximated functions (the functions with lower smoothness order). Three cases are considered: a function with a Lipschitz continuous first-order derivative, a continuous function and a Lipschitz continuous function, respectively.

Keywords

Convergence; Wu-Schaback’s

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 LDf and discussed the convergence and shape preserving properties of this operator. In their convergence theorem (theorem A in our paper), they claimed interpolated functions f(x)∈C2 . 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 LDf 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]. We assume that there are finite scattered points imagein the bounded interval [a,b] as follows:

image

and the maximum spacing is defined as

image

For image, we define its norm as

imageand its modulus of continuity as

image

The basis functions used in this paper are

image

image

where c>0 is a positive shape parameter

In 1994, Wu and Schaback proposed the quasi-interpolation operator LD:

image

where

image

They got the error estimate of this operator as follows:

Theorem A: For imagethe quasi-interpolant LD f satisfies an error estimate of type

image

where positive constants K1,K2,K3 are independent of h and c.

In 1992, Beatson and Powell [4] proposed the quasi-interpolation operator LB:

image

where

image

image

They proved the following result:

Theorem B: In interval [a,b], the error function imagesatisfies the bound

image

Meanwhile, in [4] the quasi-interpolation operator LC was defined as follows:

image

where

image

image

They got the following theorem:

Theorem C: If f has a Lipschitz continuous first-order derivative, then the maximum error of the quasi-interpolant LCf satisfies the bound

image

where

image

It should be noticed that in Theorem A, Wu and Schaback demanded the approximated function f∈C2[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:

image

where

image

Proof: We notice that quasi-interpolant LD f and LB f have the following relationship:

image (1)

In [4], Beatson and Powell have showed the relationship between LBf and LCf :

image(2)

where

image

For x∈[a,b] , we can easily get the two inequalities:

image(3)

Using (1), (2),(3), we can get

image

Further, due to Theorem C, we can get

image

Remark 1: Usually we choose c =O(h), then Theorem 1 is basically in accordance with Theorem A.

Further, for the approximated function f (x) with lower smoothness, we can get the following results:

Theorem 2: If is f(x) Lipschitz continuous in [a,b], then

image

where

image

Proof: Due to (3), it is obvious that

image

Finally, using Theorem B, we have

image

Remark 2: Since f (x) is Lipschitz continuous in [a,b] and imagewe have

image Then Theorem 2 can be rewrote as:

If f (x) is Lipschitz continuous in interval [a,b] , then

image

where

image

At last, for the general continuous approximated function f (x) , the following theorem of convergence is valid:

Theorem 3: If f (x) is continuous in [a,b] , and the interpolation knots are image(namely equally distributed), then we have the estimation:

image

Proof: Due to

image

using Theorem B, we have

image

Remark 3: Assuming c =O(h) in Theorem 3, we can conclude the convergence of Wu-Schaback’s quasi-interpolation operator dealing with continuous approximated functions when the interpolated knots are equally distributed.

Acknowledgements

Supported by the National Natural Science Foundation of China (No.11271041, No.61170005) and the Natural Science Foundation of Jilin Province (No.20130101062JC).

References

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