The homogeneous approximation property and localized Gabor frames

The homogeneous approximation property (HAP) has been introduced in order to describe the locality of Gabor expansions in the Hilbert space L 2 ( R d ) . In this manuscript the HAP is established for families of modulation spaces. Instead of the more recent theory of localized frames (Gröchenig in J...

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Bibliographic Details
Published in:Monatshefte für Mathematik 2016-10, Vol.181 (2), p.325-339
Main Authors: Feichtinger, Hans G., Neuhauser, Markus
Format: Article
Language:English
Subjects:
Approximation
Frames
Hilbert space
Mathematical analysis
Mathematics
Mathematics and Statistics
Modulation
Wavelet analysis
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Summary:The homogeneous approximation property (HAP) has been introduced in order to describe the locality of Gabor expansions in the Hilbert space L 2 ( R d ) . In this manuscript the HAP is established for families of modulation spaces. Instead of the more recent theory of localized frames (Gröchenig in J Fourier Anal Appl 10(2):105–132, 2004 ) which relies on Wiener pairs of Banach algebras of matrices, our approach is based on the constructive principles established in Feichtinger and Gröchenig (J Funct Anal 86:307–340, 1989 , Monatsh Math 108:129–148, 1989 ), Gröchenig (Monatsh Math 112:1–41, 1991 ), using the fact that generalized modulation spaces are coorbit spaces with respect to the Schrödinger representation of the Heisenberg group (cf. Feichtinger and Gröchenig in Wavelets—a tutorial in theory and applications, Academic Press, Boston, pp 359–397, 1992 ). For the (non-canonical) dual frames obtained constructively in this way the HAP property is verified.
ISSN:0026-9255
1436-5081
DOI:10.1007/s00605-016-0941-x