Dharm Prakash Singh | Searcharxiv

arXiv2026

A Bivariate Polynomial Problem for Matrices

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.

math.GM↗

arXiv2017

Analytical aspects of matrix interpolation problems and its applications

In this paper, the $mn$-dimensional space of tensor-product polynomials of two variables, of degree at most $(m-1)+(n-1)$, is considered. A theory of two-variate polynomials is developed by establishing the algebra and basic algebraic properties with respect to the usual addition, scalar multiplication, and a newly defined algebraic operation in the considered space. Further, the existence of the considered space is established with respect to the matrix interpolation problem (MIP), $P(i,j)=a_{ij}$ for all $1 \leq i \leq m$, $1 \leq j \leq n$, corresponds to a given matrix $(a_{ij})_{m \times n}$ in the space of $m \times n$ order real matrices. The poisedness of the MIP is proved and three formulae are presented to construct the respective polynomial in the considered space. After that, using construction formulae, a polynomial map from the space of $m \times n$ order real matrices to the considered space is defined. Some properties of the polynomial map are investigated and some isomorphic structures between the spaces are installed. It is proved that the considered space is isomorphic to the space of $m \times n$ order real matrices with respect to the algebra structure. The polynomials in the considered space with respect to the MIP's for the given matrices also preserve the geometric properties of the matrices such as transpose, symmetry, and skew-symmetry. Some examples are included to demonstrate and verify the results.

math.GM↗