The geometry of sets of parameters of wave packet frames

Czaja, Wojciech; Kutyniok, Gitta und Speegle, Darrin (January 2006): The geometry of sets of parameters of wave packet frames. In: Applied and Computational Harmonic Analysis, Vol. 20, No. 1: pp. 108-125

Abstract

We study wave packet systems WP(ψ, M); that is, countable collections of dilations, translations, and modulations of a single function ψ ∈ L2(R). The parameters of these unitary actions form a discrete subset M ⊂ R+ × R × R. We introduce ana- logues of the notion of Beurling density, adapted to the geometry of discrete subsets of R+ × R × R, and notions of lower and upper dimensions associated with these densities. Our goal is to describe completeness properties of wave packet systems via geometric properties of the sets of their parameters. In particular, we show necessary conditions for WP(ψ, M) to be a Bessel system, and we construct multiple examples of non-standard wave packet frames with prescribed dimensions.

• Item Type: Journal article
• Keywords: Beurling density; dimension; frame; Gabor system; wavelet; wave packet system
• Faculties: Mathematics, Computer Science and Statistics > Mathematics > Bavarian Chair for Mathematical Foundations of Artificial Intelligence
• Subjects: 500 Science > 510 Mathematics
• ISSN: 10635203
• Language: English