一类广义Birkhoff插值问题的适定插值基  

Proper interpolation basis for a class of generalized Birkhoff interpolation problems

作　　者：崔凯 

机构地区：[1]沈阳师范大学数学与系统科学学院,沈阳110034

出　　处：《沈阳师范大学学报（自然科学版）》2017年第4期441-444,共4页Journal of Shenyang Normal University:Natural Science Edition

基　　金：辽宁省科技厅自然科学基金资助项目(20170540821)

摘　　要：Birkhoff插值在应用密码学,逼近论以及PDE求解等领域有着重要应用。由于微商插值条件的不连续性,使得该问题比Lagrange和Hermite插值要复杂的多。提出了基于多项式微分条件的广义Birkhoff插值格式。探究广义Birkhoff插值问题的适定插值基,使得对任意给定的型值,在该组基张成的空间中插值时总存在唯一满足插值条件的多项式。采用代数几何的方法,通过对多样性的插值条件分析,证明了当定义插值格式的关联矩阵满足较好的性质时,适定的插值基无需繁琐的计算,可以由微分插值条件直接获得。最后通过算例验证了该方法的有效性。Birkhoff interpolation has significant applications in the fields of applied cryptography,approximation theory and PDE theory,etc.The noncontinuity of derivative conditions makes Birkhoff interpolation to be more complicated than Lagrange and Hermite interpolation.A generalized Birkhoff interpolation scheme based on polynomial differential conditions is proposed.Proper interpolation basis of the generalized Birkhoff interpolation problem is studied and a unique polynomial which satisfies interpolation conditions always exists in the space spanned by the basis for any given data values.Applying the method of algebraic geometry to analyze various interpolation conditions,we prove that when the interpolation scheme defined by incidence matrix satisfies some good properties,the proper interpolation basis can be directly obtained from differential interpolation conditions,instead of tedious computations.Finally,an example is given to illustrate the effectiveness of the proposed method.

关 键 词：BIRKHOFF插值 适定插值基 关联矩阵 正则链 

分 类 号：O241.3[理学—计算数学]