Tight Frame with Hahn and Krawtchouk Polynomials of Several Variables

Abstract

Finite tight frames for polynomial subspaces are constructed using monic Hahn polynomials and Krawtchouk polynomials of several variables. Based on these polynomial frames, two methods for constructing tight frames for the Euclidean spaces are designed. With ${\mathsf r}(d,n):= \binom{n+d-1}{n}$, the first method generates, for each $m \ge n$, two families of tight frames in ${\mathbb R}^{{\mathsf r}(d,n)}$ with ${\mathsf r}(d+1,m)$ elements. The second method generates a tight frame in ${\mathbb R}^{{\mathsf r}(d,N)}$ with $1 + N \times{\mathsf r}(d+1, N)$ vectors. All frame elements are given in explicit formulas.