Necessary and Sufficient Conditions for Almost Regularity of Uniform Birkhoff Interpolation Schemes

In this article, using a combination of the necessary and sufficient conditions for the almost regularity of an interpolation scheme, we will determine all plane uniform Birkhoff schemes, when the set of interpolated nodes has n elements and the set of derivatives we are interpolating with is )} 0 , 1 ( ), 0 , 0 {( = A . For the same A we will determine all rectangular Birkhoff uniform interpolation schemes (the set of nodes has rectangular shape) and we will take note of the fact that these two results differ significantly. Two criteria – normality condition and Pólya condition – are known to be used when establishing the almost regularity (regularity) of a multidimensional polynomial interpolation general scheme. Only one sufficient condition is known for the almost regularity (regularity) of a multidimensional interpolation scheme and this applies for a more restrictive domain, namely for the schemes of type Birkhoff. Of course, for the interpolation schemes whose components are more and more limited, the number of the necessary and sufficient conditions for regularity (almost regularity) is increasing and these are more and more explicit. The two types of interpolation schemes that we will present in this article also prove this aspect. We will present two necessary and sufficient conditions for bidimensional uniform Birkhoff schemes when the set that describes the derivatives is )} 0 , 1 ( ), 0 , 0 {( = A . For the beginning we present the following notions: 1. The finite set L 2 IN ⊂ is inferior if L v u R ⊂ ) , ( for any L v u ∈ ) , ( , where ) , ( v u R = } , : ) , {( 2 v j u i IN j i ≤ ≤ ∈ . 2. A set of nodes Z is ) , ( q p rectangular or simple rectangular ( p and q are natural numbers), if it can be written in the form )} 0 , 0 : ) , {( q j p i y x Z j i ≤ ≤ ≤ ≤ = , where p x x x ,... , 1 0 are pair-wise distinctive real numbers (similarly for q y y y ,... , 1 0 ). 3. The bivariate uniform Birkhoff interpolation scheme is the triplet ) , , ( A S Z consisting of a set (of nodes) ACTA UNIVERSITATIS APULENSIS 62 Z ={ } t t t t IR y x 1 2 ) , ( = ∈ = z , an inferior set 2 IN S ⊂ and a subset A of S . The associated (uniform, bivariate) Birkhoff interpolation problem consists in determining the polynomials