The Convergence Order Of The Approximation Operator

In the paper, we study the Triangle Interpolation approximation of the double function in chapter 3,4,and 5. Since the double trigonometric Lagrange interpolationpolynomial does not converge each continuous function f(x,y) uniformly, Weconstruct a summation factor in order to improve its convergence order .such that the integral operator with the summation factor convergence uniformly on total plane forany f(x,y) C( ) ,and has the best approximation order in chapter 3 and chapter4 ,we construct a combinational operator of the double triangle interpolation polynomial based on two kinds of nodes and study its convergence quality for the double continuous periodic functions and estimate its convergence order and so on.In chapter5 In the function is summed by means of symmetry. Thetrigomometric interpolation combination polynomial Cn(f;t,x) is constructed. If the function f{x) C2, then Cn(f;t,x) converge the function f(x) on uniformly, and the convergence order ofCn(f;t,x) is the best if , where ,t is an arbitrary add natural number.Chapter 6 is about summability theory and method of Fourier series.construct a summation factor in order to improve its convergence order,such that the integraloperator with the summation factor converges uniformly on forany ,and has the best approximation order if . In the end, chapter7 On Rogosinski type sums of Neumann-Bessel series.In this paper , we use the partial sums of Neumann-Bessel series Kn(N,B)(f;z) and construct a newoperator Hn(B,N)(f;z). Such that the new operator Hn(N,B)(f;z) converges to anycontinuous function f(z) on the unite circle |Z|=1 uniformly and has the best approximation order for f(Z) on |Z|=1.Chapter 1 introductionIn this chapter ,we introduce the meaning of the study digital approximation Theory,and the study's case of digital approximation in China and overseas,and the Main contents of this paper.Chapter 2 Basic KnowledgeIn this paper ,first, we introduce the Lagrange interpolation polynomial's definition.property and main conclusion.Then,we introduce the basic dsfinition of the best approximation polynomial and continuous and double approximation theory so on.Chapter 3 On linear summability of Double Trigonometric Interpolation polynomialsLet ,and C () be the continuous function space of the periodic function with period 2 .if the function f(x,y)C( ),then the partial sum of the double Fourier series of f(x,y) is given by Snm(f;x,y) coskxcosly +bkJ sinkxcosly +ck,l cosfx sinly + dk,l sinkx sin ly)where ak,l,bk,l, ck,l,dk,l are coefficients of the double Fourier series: Let the value of Snm(f;x,y) is equal to f(x,y) at nodes as follows:then where M=2m+1, N=2n+1 (1) (2) where ,,and satisfies the fowllowing equation:Therefore,we obtain a new double trigonometric interpolation polynomial (3) On Tnm(f;x,y)we have the following results: Theoreml:Letis valid uniformly on total plane. Theorem2: Let f(x,y) C (),thenwhere E*nm (f) is the minimum deviation with the trigonometric polynomial of H*nm(f) to approximate the function f(x,y).Chapter 4 On convergence order of Double Trigonometriclnterpolation polynomialsIn this paper .double trigonometric interpolation polynomials are constructed.To enable the arbitrary continuous periodic function f(x,y),with period 2 .convergeon the f(x,y) uniformly on total plane. And has best approximation order. Let f(x,y) C() andThen we get a double trigonometric Lagrange interpolation polynomial:(1)Double combination trigonometric interpolation polynomial Tm (f;x,y) isGiven by: Let:then Where r1 and r2 are odd natural numbersOn .we have the following results:Theoreml: Let function f(x,y) s,r () , then:Where O is independent of m and n, Enm (f ) is the minimum deviation with thetrigonometric polynomial of H*nm(f) to approximate the function f(x,y).Theorem 2: Let function f(x,y) C( ),then lim Tnm(f;x,y)= f{x,y) is valid uniformly on total plane.Chapter 5 Combinational Triangle Interpola...