School of Mathematics, Jilin University, Changchun, 130012, P.R. China
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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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.
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  in 1968, and Franke  showed they performed well in many calculations including the numerical experiments. Powell , Beatson and Powell , and Beatson and Dyn  successively proposed a number of quasiinterpolation schemes and discussed the convergence of the schemes. In 1994, Wu and Schaback  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 , and Ma and Wu  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 , and our method differs from that in . We assume that there are finite scattered points in the bounded interval [a,b] as follows:
and the maximum spacing is defined as
For , 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 LD:
They got the error estimate of this operator as follows:
Theorem A: For the quasi-interpolant LD f satisfies an error estimate of type
where positive constants K1,K2,K3 are independent of h and c.
In 1992, Beatson and Powell  proposed the quasi-interpolation operator LB:
They proved the following result:
Theorem B: In interval [a,b], the error function satisfies the bound
Meanwhile, in  the quasi-interpolation operator LC 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 LCf satisfies the bound
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:
Proof: We notice that quasi-interpolant LD f and LB f have the following relationship:
In , Beatson and Powell have showed the relationship between LBf and LCf :
For x∈[a,b] , we can easily get the two inequalities:
Using (1), (2),(3), we can get
Further, due to Theorem C, we can get
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
Proof: Due to (3), it is obvious that
Finally, using Theorem B, we have
Remark 2: Since f (x) is Lipschitz continuous in [a,b] and we have
Then Theorem 2 can be rewrote as:
If f (x) is Lipschitz continuous in interval [a,b] , then
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 (namely equally distributed), then we have the estimation:
Proof: Due to
using Theorem B, we have
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.
Supported by the National Natural Science Foundation of China (No.11271041, No.61170005) and the Natural Science Foundation of Jilin Province (No.20130101062JC).
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