Full-spark frames arising from one-parameter groups

Abstract

The present paper provides a procedure for constructing full-spark Parseval frames arising from the linear action of a 1-parameter group acting in ${\mathbb{R}}^n$. Precisely, given a square matrix A of order n, with real entries, we say that A induces the full spark frame property if the following conditions hold. There exists a vector v for which given any finite set $X\subset \mathbb{R}$ of cardinality n, the collection $\{\exp (xA)v:x\in X\}$ is a basis for ${\mathbb{R}}^n$. First, we show that if the spectrum of A is not a subset of the reals, then A does not induce the full spark frame property. Secondly, we establish that if the spectrum of A is a subset of the reals, then A induces the full spark frame property if and only if every eigenvalue of A has geometric multiplicity one. The proof of the latter fact gives a new algorithm for the construction of a class of full spark Parseval frames for ${\mathbb{R}}^n$.

Keywords:

• full spark frames
• frames

Acknowledgments

We thank the anonymous reviewer for careful and thoughtful feedback on the paper. The changes made greatly improved the quality of the presentation.

Disclosure statement

No potential conflict of interest was reported by the author(s).