The Approximation Of Some Trigonometric Interpolation Polynomials

In this paper,the approxition and saturtion about some of (0,p(D)) trigonnometric interpolation polynomials are discussedAt the same time,theapproximation properties of a (bivariate) modiffying Lagrange trisngle interpolation polynomial are studied.The paper is divied intD three chapters.the contents are following:In chapter l,the approximation and saturatian about a kind of (0, P(D)) trigonometric interpolation polnomials in the spscC2 is setted by abbsorbing theidea of (0,m) trigonometric interpolation polnomials,and the following theorem is obtained:Theoreml.1.1 If f(x)EiC2a , p(D)is an odd polynomial aboutD , then, the necessary and sufficients condition of||/, //L - 0(11 nm) is / (m-1) and f(m^l) belong to Lipl.Simultaneously,absorbing the idea of the reference[19]and based on paper [20] and [21],the approximation and saturation theories of the (0, p(D)) trigonometricinterlation polynomials in the spaces W'H " are obtained, that is:Theoreml.2.1 If p(D)is an odd polynomial aboutD , / EWm^lH" ,then, the necessary and sufficients condition of - \a m₁ = 0(1 /n1"") is f(m^l) and7(m-1} belong to Lipl. Finally, the connected questions of the (0, /)) trigonometric interpolationpolynomials in the spaces WrpHa are researched,and the folowing theorem isobtained.Theoreml.3.1 If /WEand / EWH" (mAT,l< p < oo ) then,the necessary and sufficients condition ofIn chapter 2,in order to improve the converge properties and converge order of Lagrange trigonomtric interpolation polynomial 5 ,a point (or two points) in every line's modified way is used, and the questions of S.N.Bernstein is discussed ,and thefollowing conclusion is drawed.Theorem2. 1 . 1 If f(x) e C2lc ,then,lim T (/; r, x) = f(x) is valid on tatal real axis uniformly.noTheorem2.1.2If/(jc)C2,r, ir then,where O is independent of x, , w(/(0 , 5, is the modulus of continuity of /(/) (x) , En (/) is the minimum deviation with Tn (/; r, x) to approximate the functionTheorem2.1.3 For arbitary continuous function class ,the highest convergence order of Tn(f;r,x)is .Theorem2.2.1 For arbitary /(x) e C2n ,then,limHn(f',r,x) - /(x)is vaild on tatal real axis uniformly.naoTheorem2.2.2 If /() e C2yr , (7 = 1A-, r - 1) , then ,where O is independent of x, n, /,/0) , w(/o) ,') is the modulus of continuity ofTheorem2.2.3 If f(x) e C2r, , then ,where O is independent of x, ,/,, /(r) , w(/O) , ) is the modulus of continuity ofTheorem2.2.4 If f(x) e C , and f(r} (x) e Lioa,Q \6l,82') is the modulus of continuity of fp}(x,y),E'm(f)i the mininum deviation with Tm(f\r,x,y)to approximate the function / (x, y).Theorem3.2.3 If f(x,y) C1, and f (x,y) (=Lipa,( < a < 1), then,where O is independent of n, x, y, /,..., /).Inaddition ,a bivarite trigonometric interpolation polynomials is constructed throughing average of the fundamental function and its converge properties is studied ;we have.eorem3.3.1 If f(x,y)ECr (A) , s x a,r a 6, then,here Ois independent ofn,m,x,y,f,...,fy+r),w(fx(ll),6l,0) is the modulus of continuity...