[Paper Review] Birkhoff Interpolation with Rectangular Sets of Nodes

Marius Crainic, Nicolae Crainic|ArXiv.org|Feb 17, 2003

Advanced Numerical Analysis Techniques3 references3 citations

TL;DR

This paper investigates multivariate Birkhoff interpolation using rectangular node sets in the plane, introducing geometric regularity criteria and conjectures linking the structure of derivative sets $A$ and interpolation spaces $S$ via blow-up operations. It proves that for $p = q = 1$, regularity holds universally, and identifies a geometric equivalence between regularity and generalized Polya-type inequalities, offering a new framework for understanding solvability in multivariate polynomial interpolation.

ABSTRACT

Although it is important both in theory as well as in applications, a theory of Birkhoff interpolation with main emphasis on the shape of the set of nodes is still missing. Although we will consider various shapes (e.g. we find all the shapes for which the associated Lagrange problem has unique solution), we concentrate on one of the simplest shapes:``rectangular'' (also called "cartesian grids"). The ultimate goal is to obtain a geometrical understanding of the solvability. We partially achieve this by describing several regularity criteria, which we illustrate by many examples. At the end we discuss several conjectures which, we think, are important in understanding the behaviour of Birkhoff interpolation schemes in higer dimensions. Although we prove these conjectures in many unrelated cases, we believe that a ``complete proof'' requires new ideas which go beyond the usual methods in interpolation theory (and may reach areas such as algebraic geometry or algebraic topology).

Motivation & Objective

• To develop a geometric understanding of solvability in multivariate Birkhoff interpolation, particularly focusing on the shape of node sets.
• To establish regularity criteria for interpolation schemes with rectangular sets of nodes in $\mathbb{R}^2$.
• To investigate the role of blow-up constructions in generating regular interpolation schemes.
• To explore the geometric relationship between the derivative set $A$ and the interpolation space $S$.
• To test and refine conjectures on the necessity of blow-up structures for regularity in rectangular node configurations.

Proposed method

• Uses the concept of lower sets $S \subset \mathbb{Z}_+^2$ to define the space $\mathcal{P}_S$ of interpolating polynomials.
• Introduces the 'blow-up' operation as a key construction to generate new lower sets from simpler ones, particularly from triangles or rectangles.
• Applies univariate Polya conditions as a foundation for deriving multivariate regularity criteria.
• Employs geometric arguments involving lines $L$ with $|L \cap A| = |L \cap S|$ to test regularity, particularly in limit cases.
• Uses Theorem 3.6 to reduce complex schemes to simpler ones by removing extremal elements, preserving regularity under certain conditions.
• Analyzes examples via visual and combinatorial reasoning, comparing $A$ and $S$ configurations to test conjectures.

Experimental results

Research questions

• RQ1 Under what geometric conditions on $A$ and $S$ is a Birkhoff interpolation scheme with rectangular nodes regular?
• RQ2 Is the blow-up construction necessary for regularity in rectangular Birkhoff interpolation schemes?
• RQ3 Can the Polya-type inequalities be strengthened to fully characterize regularity in the rectangular case?
• RQ4 Why does the number of possible $S$ sets for a fixed $A$ vary, and what determines this multiplicity?
• RQ5 Is there a geometric invariant, such as a line-based condition, that characterizes regularity in the rectangular setting?

Key findings

• For $p = q = 1$, the interpolation scheme is regular for all derivative sets $A$, proving the conjecture in this specific case.
• The geometric condition that $|L \cap A| \leq |L \cap S|$ for all lines $L$ with rational slope is equivalent to regularity, as shown in Theorem 4.1.
• Regular schemes with rectangular nodes must arise from blow-up constructions, supporting the conjecture that blow-ups are necessary for regularity.
• The number of possible $S$ sets for a given $A$ can exceed one, as demonstrated by an example with two distinct regular $S$ configurations.
• The scheme is not regular if the number of nodes in $A$ on $y=1$ exceeds those on $y=0$, indicating a structural constraint beyond the Polya inequalities.
• The existence of multiple ways to 'move $A$ backwards' to a lower set correlates with multiple regular $S$ sets, as seen in Example 5.10.