### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 019, 19 pages      arXiv:1309.7526      https://doi.org/10.3842/SIGMA.2014.019

### Tight Frame with Hahn and Krawtchouk Polynomials of Several Variables

Yuan Xu
Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222, USA

Received November 06, 2013, in final form February 25, 2014; Published online March 03, 2014

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.

Key words: Jacobi polynomials; simplex; Hahn polynomials; Krawtchouk polynomials; several variables; tight frame.

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