Abstract
The homogeneous approximation property (HAP) has been introduced in order to describe the locality of Gabor expansions in the Hilbert space \(L^{2}({\mathbb {R}}^{d})\). 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
The atoms \(g_{\lambda }\) are centered at \(\lambda \) (e. g. \(g_{\lambda }=\pi (\lambda )g\) for some nice g); the family \(\left( \widetilde{g}_{\lambda }\right) _{\lambda \in \Lambda }\) need not be the “canonical dual frame” in our discussion.
References
Aceska, R.: Functions of variable bandwidth: a time-frequency approach. Ph.D. thesis, Vienna (2009)
Aceska, R., Feichtinger, H.G.: Functions of variable bandwidth via time-frequency analysis tools. J. Math. Anal. Appl. 382(1), 275–289 (2011)
Feichtinger, H.G.: Modulation spaces of locally compact Abelian groups. In: Radha, R., Krishna, M., Thangavelu, S. (eds.) Proceeding of International Conference on Wavelets and Applications, January 2002, pp. 1–56. New Delhi Allied Publishers, Chennai (2003)
Feichtinger, H.G.: Modulation spaces: looking back and ahead. Sample Theory Signal Image Process 5(2), 109–140 (2006)
Feichtinger, H.G., Gröchenig, K.: Banach spaces related to integrable group representations and their atomic decompositions I. J. Funct. Anal. 86, 307–340 (1989)
Feichtinger, H.G., Gröchenig, K.: Banach spaces related to integrable group representations and their atomic decompositions II. Monatsh. Math. 108, 129–148 (1989)
Feichtinger, H.G., Gröchenig, K.: Irregular sampling theorems and series expansions of band-limited functions. J. Math. Anal. Appl. 167(2), 530–556 (1992)
Feichtinger, H.G., Gröchenig, K.: Gabor wavelets and the Heisenberg group: Gabor expansions and short time Fourier transform from the group theoretical point of view. In: Chui, C. (ed.), Wavelets–A Tutorial in Theory and Applications, vol. 2 of Wavelet Anal. Appl., pp. 359–397. Academic Press, Boston (1992)
Feichtinger, H.G., Zimmermann, G.: A Banach space of test functions for Gabor analysis. In: Feichtinger, H., Strohmer, T. (eds.) Gabor analysis and Algorithms: Theory and Applications, Applied and Numerical Harmonic Analysis, pp. 123–170. Birkhäuser, Boston (1998)
Gröchenig, K.: Describing functions: atomic decompositions versus frames. Monatsh. Math. 112, 1–41 (1991)
Gröchenig, K.: Localization of frames, Banach frames, and the invertibility of the frame operator. J. Fourier Anal. Appl. 10(2), 105–132 (2004)
Gröchenig, K.: Foundations of Time-Frequency Analysis. Appl. Numer. Harmon. Anal. Birkhäuser, Boston (2001)
Gröchenig, K.: The homogeneous approximation property and the comparison theorem for coherent frames. Sample Theory Signal Image Process 7(3), 271–279 (2008)
Heil, C.: The density theorem and the homogeneous approximation property for Gabor frames. In: Jorgensen, P.E.T., Merrill, K.D., Packer, J.D. (eds.) Representations, Wavelets, and Frames, pp. 71–102. Birkhäuser, Boston (2008)
Heil, C., Kutyniok, G.: The homogeneous approximation property for wavelet frames and Schauder bases. J. Approx. Theory 147(1), 28–46 (2007)
Hogan, J.A., Lakey, J.D.: Time-Frequency and Time-Scale Methods. Adaptive Decompositions, Uncertainty Principles, and Sampling. Birkhäuser, Boston (2005)
Hrycak, T., Das, S., Matz, G., Feichtinger, H.G.: Practical estimation of rapidly varying channels for OFDM systems. IEEE Trans. Commun. 59(11), 3040–3048 (2011)
Ramanathan, J., Steger, T.: Incompleteness of sparse coherent states. Appl. Comput. Harmon. Anal. 2(2), 148–153 (1995)
Sun, W., Liu, B.: Homogeneous approximation property for continuous wavelet transforms. J. Approx. Theory 155(2), 111–124 (2007)
Tauböck, G., Hampejs, M., Matz, G., Hlawatsch, F., Gröchenig, K.: LSQR-based ICI equalization for multicarrier communications in strongly dispersive and highly mobile environments. In: Proceedings of IEEE SPAWC 07: Signal Processing Advances in Wireless Communications, 8th Workshop on, pp. 1–5, Helsinki, 2007
Acknowledgments
The authors were supported by the Marie-Curie Excellence Grant MEXT-CT 2004-517154 (EUCETIFA). The authors would like to thank the reviewers for their comments which have helped to essentially improve the presentation of our results.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Constantin.
Rights and permissions
About this article
Cite this article
Feichtinger, H.G., Neuhauser, M. The homogeneous approximation property and localized Gabor frames. Monatsh Math 181, 325–339 (2016). https://doi.org/10.1007/s00605-016-0941-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-016-0941-x