A Bivariate Polynomial Problem for Matrices

TL;DR

This paper introduces a new bivariate polynomial problem for matrices that provides conditions for isomorphisms between matrix spaces and polynomial subspaces, with solutions and applications to interpolation problems.

Contribution

It establishes the existence, uniqueness, and construction methods for polynomials solving the problem within finite-dimensional BVPSs, linking it to bivariate interpolation.

Findings

Existence of unique solutions in new BVPS classes

Formulas for polynomial construction in BVPSs

Application to bivariate interpolation schemes

Abstract

This article proposes a bivariate polynomial problem for finite-order real matrices that endows a \textit{`sufficient condition'} for a map from the standard vector spaces of finite-order real matrices to the same dimensional bivariate polynomial subspaces (BVPSs) to be an isomorphism in some finite-dimensional BVPSs. In the process of solving, the article deals with the existence, uniqueness, and construction of the polynomials in some finite-dimensional BVPSs concerning the solution of the proposed problem. To this end, a relationship is established between the proposed problem and a class of Lagrange bivariate polynomial interpolation problems (LBVPIPs). As a result, the existence of a standard and a new class of finite-dimensional BVPSs of various total degrees has been established in which the proposed problem always possesses a unique solution. In addition, some formulas are derived to construct the needed polynomials in these BVPSs. Further, the possible applicability of the proposed problem is discussed in LBVPIPs, focusing on the finite rectangular schemes of bivariate interpolation points on the natural Cartesian grid. At last, some numerical examples are considered to justify the theoretical findings.