The Simultaneous Approximation Based On Birkhoff Interpolation Polynomials

Studies of mathematical inequalities play a significant role in the fields of pure mathematics and applied mathematics and now have formed a systematic science theory.Inequalities involving functions and their derivatives are the most noticeable problems in the area of function approx-imation theory,among which the well-known inequalities include Landau-Kolmogorov inequality,Gorny inequality,Wirtinger inequality,Picone inequality,Schmidt inequality,Sobolev inequality,Bernstein inequality,Markov inequality,et al.At present,there are a large number of research results about Wirtinger-type inequality and generalized forms of its higher order derivative–Picone-type inequality.These problems are often related to the simultaneous approximation of interpo-lation operators.It is noted that the simultaneous approximation problems studied in the past were concerned with the simultaneous approximation of Hermite interpolation operator,where the information values at the continuous integers interpolation nodes.Birkhoff interpolation as a gen-eralization of Hermite interpolation,which mainly addressed the problems that the information values at the non-continuous integers interpolation nodes.This article is devoted to investigating the simultaneous approximation errors of Birkhoff interpolation and to obtaining the corresponding exact inequalities of Wirtinger-type and so on.In the first chapter,in combination with the references,we introduce the history of researches,generalized formulas and the current research status at home and abroad about Wirtinger-type inequality and give the structure of this paper.In the second chapter,introducing the basic concepts and the sectional derivation processes of some lemmas related to this subject.It consists mainly of the remainder theorem of Birkhoff interpolation polynomial and the eigenvalue theory of Hilbert-Schmidt operator.We also show the extended formula and the theoretical proof of(?)In the third chapter,according to the results and corollaries obtained in the second chapter,we give the simultaneous approximation of Birkhoff interpolation under certain interpolation sit-uations.Providing numerical calculations to illustrate our theorems and giving the corresponding exact inequalities.