OPTIMAL BIRKHOFF INTERPOLATION AND BIRKHOFF NUMBERS IN SOME FUNCTION SPACES

This paper investigates the optimal Birkhoff interpolation and Birkhoff numbers of some function spaces in space L∞[-1,1]and weighted spaces Lp,ω[-1,1],1≤p<∞,with w being a continuous integrable weight function in(-1,1).We proved that the Lagrange interpolation algorithms based on the zeros of some polynomials are optimal.We also show that the Lagrange interpolation algorithms based on the zeros of some polynomials are optimal when the function values of the two endpoints are included in the interpolation systems.

HERMITE—BIRKHOFF INTERPOLATION OF SCATTERED DATA BY RADIAL BASIS FUNCTIONS

For Hermite-Birkhoff interpolation of scattered multidumensional data by radial basis function (?),existence and characterization theorems and a variational principle are proved. Examples include (?)(r)=r^b,Duchon's thin-plate splines,Hardy's multiquadrics,and inverse multiquadrics.

Multivariate Vandermonde Determinants and General Birkhoff Interpolation

In this paper, we consider the Straight Line Type Node Configuration C （SLTNCC） in multivariate polynomial interpolation as the result of different kinds of transformations of lines （such as parallel translations, rotations）. Corresponding to these transformations we define different kinds of interpolation problems for the SLTNCC. The expression of the confluent multivariate Vandermonde determinant of the coefficient matrix for each of these interpolation problems is obtained, and from this expression we conclude the related interpolation problem is unisolvent. Also, we give a kind of generalization of the SLTNCC in Section 5. As well, we obtain an expression of the interpolating polynomial for a kind of interpolation problem discussed in this paper.

ON BIRKHOFF INTERPOLATION (0;2) CASE

P. Turan and his associates considered in detail the problem of (0.2) interpolation based on the zeros of πn(x). Motivated by these results and an earlier result of Szabados and Varma[9] here we consider the problem of existence, uniqueness and explicit representation of the interpolatory polynomial Rn (x) satisfying the function values at one set of nodes and the second derivative on the other set of nodes. It is important to note that this problem has a unique solution provided these two sets of nodes are chosen properly. We also promise to have an interesting convergence theorem in the second paper of this series, which will provide a solution to the related open problem of P. Turan.

On the Unique Minimal Monomial Basis of Birkhoff Interpolation Problem

This paper studies the minimal monomial basis of the n-variable Birkhoff interpolation problem. First, the authors give a fast B-Lex algorithm which has an explicit geometric interpretation to compute the minimal monomial interpolation basis under lexieographie order and the algorithm is in fact a generalization of lex game algorithm. In practice, people usually desire the lowest degree interpolation polynomial, so the interpolation problems need to be solved under, for example, graded monomial order instead of lexicographie order. However, there barely exist fast algorithms for the non- lexicographic order problem. Hence, the authors in addition provide a criterion to determine whether an n-variable Birkhoff interpolation problem has unique minimal monomial basis, which means it owns the same minimal monomial basis w.r.t, arbitrary monomial order. Thus, for problems in this case, the authors can easily get the minimal monomial basis with little computation cost w.r.t, arbitrary monomial order by using our fast B-Lex algorithm.

On Two Birkhoff-Type Interpolations with First- and Second-Order Derivative

In this paper, we consider two interpolations of Birkhoff-type with integer-order derivative. The Birkhoff interpolation is related with collocation method for the corresponding initial or boundary value problems of differential equations. The solvability of the interpolation problems is proved. For Gauss-type interpolating points, error of interpolation approximation is deduced. Also, we give efficient algorithms to implement the concerned interpolations.

Regularity of (0,2)-interpolation on the Zeros of the Lascenov Polynomials

In is paper, a necessary and sufficient condition of regularity of (0,2)_interpolation on the zeros of the Lascenov Polynomials R (α,β) n(x)(α,β>-1) in a manageable form is estabished. Meanwhile, the explicit representation of the fundamental polynomials, when they exist, is given.

ON HIGH DIMENSIONAL INTERPOLATION

This paper introduces the definition of the Orthogonal Type Node Configuration and discusses the corresponding multivariate Lagrange, Hermite and Birkhoff interpolation problems in high dimensional space R s(s>2). This node configuration can be considered to be a kind of extension of the Cross Type Node Configuration , in R 2 to high dimensional spaces. And the Mixed Type Node Configuration in R s(s>2) is also discussed in this paper in an example.

A Kind of Generalization of the Curve Type Node Configuration in R^S（S 〉 2）

A kind of generalization of the Curve Type Node Configuration is given in this paper, and it is called the generalized node configuration CTNCB in Rs(s > 2). The related multivariate polynomial interpolation problem is discussed. It is proved that the CTNCB is an appropriate node configuration for the polynomial space Pns(s > 2). And the expressions of the multivariate Vandermonde determinants that are related to the Odd Curve Type Node Configuration in R2 are also obtained.

New Rational Interpolation Basis Functions on the Unbounded Intervals and Their Applications

Based on the rational system of Legendre rational functions,we construct two set of new interpolation basis functions on the unbounded intervals.Their explicit expressions are derived,and fast and stable algorithms are provided for computing the new basis functions.As applications,new rational collocation methods based on these new basis functions are proposed for solving various second-order differential equations on the unbounded domains.Numerical experiments illustrate that our new methods are more effective and stable than the existing collocation methods.

THE ELLIPTIC TYPE NODE CONFIGURATION AND INTERPOLATION IN R^2

In this paper, we have obtained an expression of the bivariate Vandermonde determinant for the Elliptic Type Node Configuration in R-2, and discussed the possibility of the corresponding multivariate Lagrange, Hermite and Birkhoff interpolation.

REGULARITY AND EXPLICIT REPRESENTATION OF（0，1，…，m－2，m） INTERPOLATION ON THE ZEROS OF（1－x^2）P_（n－2）^（α，β）（x）

Necessary and sufficient conditions for the regularity and q-regularity of (0,1,…,m-2,m)interpolation on the zeros of (1-x2)P_(n-2)￣(α,β)(x) (α,β>-1) in a manageable form are established,where P_(n-2)￣(α,β)(x) stands for the (n-2)th Jacobi polynomial. Meanwhile, the explicit representation of the fundamental polynomials, when they exist, is given. Moreover, we show that under a mild assumption if the problem of (0,1,…,m-2,m) interpolation has an infinity of solutions then the general form of the solutions is fo(x)+Cf(x) with an arbitrary constant C.