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A Research Of Approximation Scheme Based On Two-variable Box Spline

Posted on:2018-01-21Degree:MasterType:Thesis
Country:ChinaCandidate:D D JiaFull Text:PDF
GTID:2310330515483081Subject:Computational Mathematics
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Approximation theory is an important branch in the field of computational mathemat-ics,which is significant to both theoretical research and practical application.At the same time,approximation theory plays an important role in algebra,differential equations and oth-er mathematical problems.We discuss two variables case of approximation theory.we know that Box spline is the generalization of B spline and Box spline is a very important basis function in the multivariate approximation problem.In this paper,we extend the approxi?mation scheme based on B spline[1]to the case of two variables,that is,we construct the approximation operator which is based on two-variable Box spline function to approximate two-variable functions.At the same time,the derivative of two-variable functions can be approximated by taking the derivative of the approximation operator.In order to obtain the interpolation operator with algebraic accuracy,we construct two-variable approximation op-erator in two stages:In the first stage,we construct an interpolation operator L based on two-variable Box spline(without algebraic accuracy)and a quasi-interpolation operator with algebraic accura-cy Q based on two-variable Box spline respectively;In the second stage,a new approximation operator l is constructed by making boolean sum of the interpolation operator L and quasi-interpolation operator Q.The operator l has not only the same algebraic accuracy as the quasi-interpolation operator Q but also the inter-polation property.And then,we also can approximate to the derivative of two-variable functions by taking derivative of the approximation operator l.In this paper,we construct the approximation operator based on product-type Box spline and(2,2,2)order Box spline respectively.In the fourth chapter of this paper,we use different types of two-variable functions as approximated functions and show the approximation property of l by numerical experiments.This paper is divided into five chapters:The first chapter is the introduction of this paper:in the first section,we mainly discuss the origin?development of approximation theory,In the second section,we introduce the main idea of this paper.In the second chapter,we introduce the definition of Box spline from two aspects in the first section.In the second section,we introduce several important properties of Box spline including the degree of Box spline,smoothness,support property,non negative unit decom-position and central symmetry property.The third chapter is the core of this paper.In the first section,we introduce the approx-imation operator based on B spline[1].In the second section,we introduce an general form of approximation operator based on Box spline.In the last two sections,we give specific form of approximation operator lm based on product-type Box spline and(2,2,2)order Box spline N(2,2,2)(x).In the fourth chapter,the numerical experiments,we discuss approximation effect of l(2,2,2)using different kinds of two-variable functions as approximation functions.At first,we use f1=xy,f2 ?X2y,f3 = xy2 as the approximated functions.we show that l(2,2,2)is an accurate approximation operator for f1=xy,f2-=x2y,f3=xy2 by comparing the image of the operator l(2,2,2)and the image of the original function,that is,the algebraic accuracy of l(2,2.2)is 3.For functions which are not approximated by l(2,2,2)precisely,we change the interpolation nodes step h.The approximation capability of l(2,2,2)is further illustrated by observing the variation of approximation error with h.The fifth chapter is the conclusion of this paper,which summarizes the main idea of this paper and puts forward the shortcomings of this paper.
Keywords/Search Tags:Box spline, Interpolation, Quasi-interpolation, Algebraic accuracy
PDF Full Text Request
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