李艷坡,何尚琴,高明晶,鄭國萍,郭亞軍

(河北科技師范學(xué)院數(shù)學(xué)與信息科技學(xué)院,河北 秦皇島 066004)

在等距節(jié)點(diǎn)處的反周期函數(shù)的一類三角插值問題

李艷坡,何尚琴,高明晶,鄭國萍,郭亞軍

(河北科技師范學(xué)院數(shù)學(xué)與信息科技學(xué)院,河北 秦皇島 066004)

研究了以π為周期的反周期函數(shù)的一類缺項(xiàng)三角插值,解決了在等距節(jié)點(diǎn)處的反周期函數(shù)的(0,P(D))三角插值問題,得到了解存在的條件、插值函數(shù)的顯式表達(dá)式及其收斂階.

反周期函數(shù);插值;正則;收斂階

1 幾個(gè)引理

2 主要結(jié)果

3 主要結(jié)果的證明

4 例題

當(dāng)m為奇數(shù)時(shí),Po(i(2 j-1))+Po(i(2n-2 j+1))=(2 j-1)m+(2n-2 j+1)m≠0.

當(dāng)m為偶數(shù)時(shí),Pe(i(2 j-1))-Pe(i(2n-2 j+1))=(2 j-1)m-(2n-2 j+1)m≠0.由此可見,文獻(xiàn)[7]的結(jié)論是本文的特殊情況.

例2 令P(t)=t+t3,即P(t)是奇多項(xiàng)式,則當(dāng)

Po(i(2j-1))+Po(i(2n-2 j+1))=(2 j-1)-(2 j-1)3+(2n-2 j+1)-(2n-2 j+1)3≠0時(shí),定理1-3總成立.

令P(t)=t2+t4,即P(t)是偶多項(xiàng)式,則當(dāng)

Pe(i(2 j-1))-Pe(i(2n-2 j+1))=-(2j-1)2+(2 j-1)4-(-(2n-2j+1)2+(2n-2j+1)4)≠0時(shí),定理1-3總成立.

[1]LOREN TZ G G.Birkhoff Interpo lation[M].London:A ddison-W esley Publishing Com pany,1983.

[2]SHARMA A,XU Y.Mean convergence of trigonometric interpolationson equidistant nodes:Birkhoff dada[J].Bull Polish Acard Sci,1991,39(3):199-206.

[3]SHARMA A,SZABADOSJ,VARGA R S.Some 2-periodic trigonometric interpolationsof equidistant nodes[J].Analysis,1991,11:165-190.

[4]L IU Yongping.The app roximation properties of certain trigonometric interpolation polynomia operato res[J].No rthestern Math,1988,4(3):289-308.

[5]DELVOS F J,KNOCHE L.Lacunary interpolation by antiperiodic trigonometric polynomials[J].B IT,1999,39(3):430-450.

[6]韓惠麗.Hermite半三角插值公式及其應(yīng)用[D].武漢:武漢大學(xué)數(shù)學(xué)與統(tǒng)計(jì)學(xué)院,2004.

[7]何尚琴,侯象乾.反周期函數(shù)(0,m)三角插值的收斂性[J].寧夏大學(xué)學(xué)報(bào):自然科學(xué)版,2006,27(3):218-224.

[8]謝庭藩,周頌平.實(shí)函數(shù)逼近論[M].杭州:杭州大學(xué)出版社,1998.

(責(zé)任編輯:王蘭英)

Problem of a Kind Antiperiodic Trigonometric Interpolation on Equidistant Nodes

LIYan-po,HE Shang-qin,GAOMing-jing,ZHENG Guo-ping,GUO Ya-jun
(Schoo l of M athematics and Info rmation Techno logy,Hebei No rm al University of Science and Techno logy,Qinhuangdao 066004,China)

The Birkhoff trigonometric interpolation ofπ-antiperiodic function was studied,the p roblem of theπ-an tiperiodic function(0,P(D))in terpo lation on equidistant nodes w as solved,the existing conditionsof the so lution was obtained,the exp licit fo rm and the rate of convergence were go t in the corresponding condition.

antiperiodic function;interpolation;regular;the rate of convergence

O 174.41

A

1000-1565(2010)06-0617-05

2009-12-05

河北省自然科學(xué)基金數(shù)學(xué)研究專項(xiàng)(08M 004)

李艷坡(1973-),女,河北昌黎人,河北科技師范學(xué)院講師,主要從事函數(shù)論方向研究.