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Summary: Accumulative density
Gerard Ascensia and Gitta Kutyniokb
aDepartament de Matem`atiques, Universitat Aut`onoma de Barcelona, 08193 Belleterra
(Barcelona), Spain;
bMathematical Institute, JustusLiebigUniversity Giessen, 35392 Giessen, Germany
ABSTRACT
In this paper we study a notion of density for subsets of R2
called accumulative density, which is similar to
the density for sequences in the unit disc developed by Seip. Along the way we derive some new properties of
Beurling density. Finally, we prove that the accumulative density and the Beurling density coincide.
Keywords: Accumulative density, Beurling density, Separated sequence
1. INTRODUCTION
Gabor and wavelet systems are among the most important systems used for signal processing purposes. For many
years regular Gabor systems and classical wavelet systems have been extensively employed. Recently, also the
more general irregular Gabor and wavelet systems were studied. In this context, density conditions have turned
out to be an especially useful and elegant tool. Conceptually, they are used to deliver necessary conditions for
a system to form a frame or a Riesz basis. The densities employed for Gabor systems are the lower and upper
Beurling density, whereas for wavelet systems a version of the Beurling density adapted to the geometry of the
affine group has been used.
Given a function g L2
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