The Construction Of Multi-Quadric Quasi-Interpolation Operator And Related Properties

Based on the research of radial basis interpolation and multiquadric quasi-interpolation, especially quasi-interpolation operators: LAf(x), LBf(x), Lcf(x) and LDf(x), some properties such as polynomial reproduction property, shape preservation and convexity are verified. Some images of LAf(x), LBf(x), Lcf(x) and LDf(x) are shown in the paper. The main research result in this paper is constructing the four new quasi-interpolation operators L1f(x)、L2f(x)、L3f(x) and L4f(x). By improving LDf(x) and Lcf(x) we get L1f(x) and L2f(x). At the base of L1f(x) and L2f(x), the derivative values of the endpoints are replaced by a linear combination of the divided difference, the operators L3f(x) and L4f(x) are received. They are more suitable for practical application.At the same time, the shape preservation and convexity of new operators are discussed in this paper. We draw image that the new quasi-interpolation operators approximate the function of one dimensional, respectively according to the known data, so as to the 2-D function and 3-D function, the images reflects the degree of its approximation.In particular, the numerical examples are shown that the new interpolation operators are effective.