三行H-B插值关联矩阵的正则性

doi:10.12286/jssx.1983.4.421

计算数学 ›› 1983, Vol. 5 ›› Issue (4): 421-429.DOI: 10.12286/jssx.1983.4.421

三行H-B插值关联矩阵的正则性

黄超凡

华东石油学院

出版日期:1983-04-14 发布日期:1983-04-14

PDF ( 772 ) 引用被引次数 ( 2 )

黄超凡. 三行H-B插值关联矩阵的正则性[J]. 计算数学, 1983, 5(4): 421-429.

REGULARITY OF THE THREE--ROW INCIDENCE MATRIX OF H-B INTERPOLATION PROBLEM

Huang Chao-fan

Online:1983-04-14 Published:1983-04-14

1. 引言和引理Hermite-Birkhoff插值是G.D.·Birkhoff于1906年提出的,1966年I.J.Schoenberg 引出了关联矩阵的概念后有了迅速的发展.关于它的发展概况在[9—12,15]中有详尽的论述.由于引入行的结合(coalesence)方法后,可把多于三行的H-B插值关联

In this paper we suppose that the three row incidence matrix E satisfies Polya condition.Let p, r be positive integers and let the number of non--zero entries in the first and thirdrows of E be p, r respectively. Let i_1 < i_2 < … < i_p and k_1 < k_2 < … < k_r denote thepositions of the 1's in the first and third rows respectively, and let l_1 < l_2 < … < l_(p+r).denote the positions of the o's in the middle row. Suppose further that l_(s1),…,l_(sp) denote theminimal sequence with l_(sj)≥i_j, j = 1, …, p in {l_1,…, l_(p+r)}, and that {l_(s'_1),…,l_(s'_r)} = {l_1,…, l_(p+r)}\{l_(s_1),…,l_(s_p)}. Similarly suppose that l_(t'_1),…l_(t'_r) denote the minimalsequence with l_(t'_j)≥k_j, j = 1,…, r and that {l_(t_1),…, l_(t_p)} = {l_1,…, l_(p+r)}\{l_(t'_1),…, l_(t'_r)}. We shall prove the following theorem: Theorem 1. If E satisfies Polya condition and ifthen E is strong non--poised. Conjecturc. Let E = (e_(ij)) be a three--row incidence matrix with where 0 ≤ k_1 < k_2 <…< k_(2p) ≤ n. Then E is poised, if and only if k_(2p)--k_1≡1, k_(2p-1) -- k-2 ≡1, …, k_(p+1) -- k_p ≡ 1(mod 2).

分享此文:

[1] A. Sharma, J. Tzimbalario, Some strongly non-poised H-B problems, J. Math. Anal Appl.,63． 1978,.521--524．
[2] E. Passow, Hermite-Birkhoff interpolation: A class of non-poised matrices, J. Math. Anal. Appl.,62, 1978, 140--147．
[3] R. Devore,A. Meir, A. Sharma, Strongly and weakly non-poised H-B interpolation problems,Canad. J. Math., 25(1973) , 1040--1050．
[4] G. G. Lorentz, S. S. Stangler, K. L. Zeller, Regularity of some special Birkhoff matrices, in "Ap-proximation Theory Ⅱ" (G. G. Lorentz,,C. K. Chui and L. L. Schumaker, Eds.) 405-410,Academic Press, New York, 1976．
[5] I. J. Schoenberg, On Hermite-Birkhoff interpolation, J. Math. Anal. Appl., 16(1966) , 538-543．
[6] S. Karlin, Total Positivity, Stanford Univ. Press. Stanford, Calif., 1968．
[7] D. Ferguson. The question of uniqueness for G. D. Birkhoff interpolation problems,J. Appro-ximation Theory, 2(1968) , 1--28．
[8] G. G. Lorentz, The Birkhoff interpolation problem: New methods and results, in Proceedings Int Conferonce, Oberwolfach Birkhauser Verlag, Basel 1974, 481--501．
[9] S. Karlin, J. M. Karon, Poised and non-poised Hermite-Birkhoff Interpolation,Indiana Univ.Math. J, 21(1972) ,1131-1170．
[10] A. Sharma, Some poised and non-poised problems of interpolation. SIAM Rew, 14(1972) , 129--151．
[11] G. G. Lorentz, Birkhoff Interpolation Problems, CNA-102,University of Texas, Austin,July,1975．
[12] G. G. Lorentz, S. D. Riemenschneider, Recent progress in Birkhoff interpolation,in "Approximation Theory and Functional Analysis" (J. B. Prolla Ed.) Mathematics Stuides No. 35, pp.187--236, North Holland, Amsterdam, 1979,
[13] G. G. Lorentz, K. L. Zeller, Birkhoff inte polation problem: Coalesence of rows, Arch. Math.(Basel), 26(1975) , 189-192．
[14] G. G. Lorentz,Birkhoff interpolation and the problem of free matrices. J. Approximation Theory,6(1972) . 283--290．
[15] 黄超凡,叶正麟,H-B插值问题,待发表.

No related articles found!

阅读次数

全文

摘要

三行H-B插值关联矩阵的正则性

REGULARITY OF THE THREE--ROW INCIDENCE MATRIX OF H-B INTERPOLATION PROBLEM

扫码分享