Scott N. Kersey | Searcharxiv

arXiv2017

Invertibility of Submatrices of Pascal's Matrix and Birkhoff Interpolation

The infinite upper triangular Pascal matrix is $T = [\binom{j}{i}]$ for $0\leq i,j$. It is easy to see that any leading principle square submatrix is triangular with determinant $1$, hence invertible. In this paper, we investigate the invertibility of arbitrary square submatrices $T_{r,c}$ comprised of rows $r=[r_0,\ldots,r_m]$ and columns $c=[c_0,\ldots,c_m]$ of $T$. We show that $T_{r,c}$ is invertible iff $r \leq c$ (i.e., $r_i \leq c_i$ for $i=0, \ldots, m$), or equivalently, iff all diagonal entries are nonzero. To prove this result we establish a connection between the invertibility of these submatrices and polynomial interpolation. In particular, we apply the theory of Birkhoff interpolation and \polya{} systems.

math.NA↗

arXiv2014

Dual Basis Functions in Subspaces

In this paper we study dual bases functions in subspaces. These are bases which are dual to functionals on larger linear space. Our goal is construct and derive properties of certain bases obtained from the construction, with primary focus on polynomial spaces in B-form. When they exist, our bases are always affine (not convex), and we define a symmetric configuration that converges to Lagrange polynomial bases. Because of affineness of our bases, we are able to derive certain approximation theoretic results involving quasi-interpolation and a Bernstein-type operator. In a broad sense, it is the aim of this paper to present a new way to view approximation problems in subspaces. In subsequent work, we will apply our results to dual bases in subspaces of spline and multivariate polynomial spaces, and apply this to the construction of blended function approximants used for approximation in the sum of certain tensor product spaces.

math.NA↗