Beurling Density and Shift-Invariant Weighted Irregular Gabor Systems

Kutyniok, Gitta (1. Mai 2006): Beurling Density and Shift-Invariant Weighted Irregular Gabor Systems. In: Sampling Theory in Signal and Image Processing, Bd. 5: S. 163-181

Abstract

In this paper we introduce and study a concept to assign a shift-invariant weighted Gabor system to an irregular Gabor system while preserving special properties, such as being a frame. First we extend the notion of Beurling density to weighted subsets of ℝd. We then derive a useful reinterpretation of this definition by using arbitrary piecewise continuous, positive functions in the amalgam space W(L∞, L1) to measure weighted Beurling density, thereby generalizing a result by Landau in the non-weighted situation. Using, in addition, decay properties of the short-time Fourier transform of functions contained in the modulation space M1(ℝd), we establish a fundamental relationship between the weighted Beurling density of the set of indices, the frame bounds, and the norm of the generator for weighted Gabor frames. Finally, we prove that the relation between an irregular Gabor system and its shift-invariant counterpart imposes special conditions on the weighted Beurling densities of their sets of indices.

• Dokumententyp: Zeitschriftenartikel
• Fakultät: Mathematik, Informatik und Statistik > Mathematik > Professur für Mathematische Grundlagen des Verständnisses der künstlichen Intelligenz
• Themengebiete: 500 Naturwissenschaften und Mathematik > 510 Mathematik
• ISSN: 1530-6429
• Sprache: Englisch