关于反周期函数的2-周期(0,p(D))三角插值

On the 2-periodic (0,p(D)) trigonometric interpolation of antiperiodic function

目的为了克服以2π为周期的三角插值问题所对应的插值空间Tn,ε(ε=0或1)对平移运算和求导运算不封闭,给出以π为周期的反周期函数的2-周期(0,p(D))三角插值。方法采用不同于Franz-Jurgen Delvos等人(Franz-Jurgen Delvos.BIT,1993,33(1),113-123;Franz-Jurgen Delvos,Ludger Knoche,BIT,1999,39(3):430-450.)的研究方法,通过不断求解给出结果。结果与结论给出了问题正则的充分必要条件及正则时基多项式的明显表达式,即r2v(x)=-(1/n)sum from j=1 to 2n( C2j-1cos(2j-1)(x-x2v)-D2j-1sin(2j-1)(x-x2v))/(Δ2j-1) ,q2v+1(x)=1/n sum from j=1 to n(1/Δ2j-1)[A2j-1cos(2j-1)(x-x2v+1)-iB2j-1sin(2j-1)(x-x2v+1)],其中v=0,1,…,n-1。

Aim To advoid the spaces Tn,ε（ε = 0 or 1） for 2π-periodic trigonometric interpolation problem are variant with respect to translation and differentiation, the 2-periodic （0,P（D）） trigonometric interpolations of antiperiodic function are considered. Method The methods that are different with Franz-Jurgen Delvos＇ （Franz-Jurgen Delvos. BIT, 199：3, ：3：3 （ 1 ）, 11：3-123 ; Franz-Jurgen Delvos, Ludger Knoche, BIT , 1999,39（3）. 430-450. ） are used, the results are given by resolving. Results and Conclusions When the problem is regular, the necessary and sufficient conditions are given, furthermore ,the fundamental polynomials of interpolation are given ,they are r2v（x）=-1/n^2n∑ j=1 C2j-1cos（2j-1）（x-x2v）-D2j-1sin（2j-1）（x-x2v）/Δ2j-1 ,q2v＋1（x）=1/n ^n∑ j=1 1/Δ2j-1[A2j-1cos（2j-1）（x-x2v＋1）-iB2j-1sin（2j-1）（x-x2v＋1）]where v=0,1,…,n-1.