Coorbit theory and Bergman spaces. (English) Zbl 1322.46023
Vasil’ev, Alexander (ed.), Harmonic and complex analysis and its applications. Cham: Birkhäuser/Springer (ISBN 978-3-319-01805-8/hbk; 978-3-319-01806-5/ebook). Trends in Mathematics, 231-259 (2014).
Let \(G\) be a locally compact topological group and \(\pi\) be a unitary irreducible representation of \(G\) on a Hilbert space \(H\). Let \(g \in H\) be an analyzing window (atom). Then
\[
(V_g f)(x) = (f, \pi (x) g ), \qquad x\in G, \quad f,g \in H,
\]
is called the voice transform of \(F\). Under some restrictions, one has the reproducing formula
\[
V_g f = V_g f * V_g g, \qquad f\in H.
\]
This observation can be taken as a starting point to wavelet expansions in several function spaces of Besov-Triebel-Lizorkin type in \(\mathbb R^d\), in modulation spaces, weighted Bergman spaces, etc. This survey describes the background, Section 1, and concentrates in Section 2 on weighted Bergman spaces over the unit disc.
For the entire collection see [Zbl 1278.30003].
For the entire collection see [Zbl 1278.30003].
Reviewer: Hans Triebel (Jena)
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 46E15 | Banach spaces of continuous, differentiable or analytic functions |
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
| 46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |
References:
| [1] | J. Arazy, Some aspects of the minimal, Möbius-invariant space of analytic functions on the unit disc., Interpolation spaces and allied topics in analysis, Proc. Conf., Lund/Swed. 1983, Lect. Notes Math. 1070, 24-44 (1984) · Zbl 0553.46021 |
| [2] | Arazy, J.; Fisher, S.; Peetre, J., Möbius invariant function spaces, J. Reine Angew. Math., 363, 110-145 (1985) · Zbl 0566.30042 |
| [3] | Boggiatto, P.; Cordero, E.; Gröchenig, K., Generalized anti-Wick operators with symbols in distributional Sobolev spaces, Integr. Equat. Oper. Theory, 48, 4, 427-442 (2004) · Zbl 1072.47045 · doi:10.1007/s00020-003-1244-x |
| [4] | Christensen, J. G.; Olafson, G., Examples of coorbit spaces for dual pairs, Acta Appl. Math., 107, 1-2, 25-48 (2009) · Zbl 1190.43004 · doi:10.1007/s10440-008-9390-4 |
| [5] | Christensen, J. G.; Olafsson, G., Coorbit spaces for dual pairs, Appl. Comput. Harmon. Anal., 31, 2, 303-324 (2011) · Zbl 1221.43003 · doi:10.1016/j.acha.2011.01.004 |
| [6] | Christensen, J. G.; Mayeli, A.; Olafsson, G., Coorbit description and atomic decomposition of Besov spaces, Arxiv preprint, arXiv, 1110-6676 (2011) |
| [7] | Christensen, J. G.; Mayeli, A.; Olafsson, G., Coorbit description and atomic decomposition of Besov spaces, Numer. Funct. Anal. Opt., 33, 7-9, 847-871 (2012) · Zbl 1251.30062 · doi:10.1080/01630563.2012.682134 |
| [8] | Cordero, E.; Rodino, L., Wick calculus: a time-frequency approach, Osaka J. Math., 42, 1, 43-63 (2005) · Zbl 1075.35126 |
| [9] | Dahlke, S.; Steidl, G.; Teschke, G., Coorbit spaces and Banach frames on homogeneous spaces with applications to the sphere, Adv. Comput. Math., 21, 1-2, 147-180 (2004) · Zbl 1113.42023 · doi:10.1023/B:ACOM.0000016435.42220.fa |
| [10] | Dahlke, S.; Steidl, G.; Teschke, G., Weighted coorbit spaces and Banach frames on homogeneous spaces, J. Fourier Anal. Appl., 10, 5, 507-539 (2004) · Zbl 1098.42025 · doi:10.1007/s00041-004-3055-0 |
| [11] | S. Dahlke, G. Teschke, K. Stingl, Coorbit theory, multi-α-modulation frames, and the concept of joint sparsity for medical multichannel data analysis, 2008, http://user.hs-nb.de/ teschke/ps/55.pdf · Zbl 1184.94195 |
| [12] | Dahlke, S.; Fornasier, M.; Rauhut, H.; Steidl, G.; Teschke, G., Generalized coorbit theory, Banach frames, and the relation to alpha-modulation spaces, Proc. Lond. Math. Soc., 96, 2, 464-506 (2008) · Zbl 1215.42035 · doi:10.1112/plms/pdm051 |
| [13] | Dahlke, S.; Kutyniok, G.; Steidl, G.; Teschke, G., Shearlet coorbit spaces and associated Banach frames, Appl. Comput. Harmon. Anal., 27, 2, 195-214 (2009) · Zbl 1171.42019 · doi:10.1016/j.acha.2009.02.004 |
| [14] | Dahlke, S.; Steidl, G.; Teschke, G., Shearlet coorbit spaces: Compactly supported analyzing shearlets, traces and embeddings, J. Fourier Anal. Appl., 17, 6, 1232-1255 (2011) · Zbl 1243.46023 · doi:10.1007/s00041-011-9181-6 |
| [15] | S. Dahlke, G. Steidl, G. Teschke, Multivariate shearlet transform, shearlet coorbit spaces and their structural properties, http://user.hs-nb.de/ teschke/ps/73.pdf · Zbl 1247.42025 |
| [16] | Dahlke, S.; Häuser, S.; Teschke, G., Coorbit space theory for the Toeplitz shearlet transform, Int. J. Wavelets Multiresolut. Inf. Process., 10, 4, 1250037 (2012) · Zbl 1252.42031 |
| [17] | P. Duren, A. Schuster, Bergman Spaces, vol. 100 of Mathematical Surveys and Monographs. (Amer. Math. Soc. (AMS), Providence, RI, 2004) · Zbl 1059.30001 |
| [18] | Feichtinger, H. G., On a new Segal algebra, Monatsh. Math., 92, 269-289 (1981) · Zbl 0461.43003 · doi:10.1007/BF01320058 |
| [19] | Feichtinger, H. G.; Gröchenig, K., A unified approach to atomic decompositions via integrable group representations, Lect. Note Math., 1302, 52-73 (1988) · Zbl 0658.22007 · doi:10.1007/BFb0078863 |
| [20] | Feichtinger, H. G.; Gröchenig, K., Banach spaces related to integrable group representations and their atomic decompositions, I. J. Funct. Anal., 86, 2, 307-340 (1989) · Zbl 0691.46011 · doi:10.1016/0022-1236(89)90055-4 |
| [21] | Feichtinger, H. G.; Gröchenig, K., Banach spaces related to integrable group representations and their atomic decompositions, II. Monatsh. Math., 108, 2-3, 129-148 (1989) · Zbl 0713.43004 · doi:10.1007/BF01308667 |
| [22] | Fornasier, M., Nota on the coorbit theory of alpha- modulation spaces (2005) |
| [23] | Fornasier, M.; Rauhut, H., Continuous frames, function spaces, and the discretization problem, J. Fourier Anal. Appl., 11, 3, 245-287 (2005) · Zbl 1093.42020 · doi:10.1007/s00041-005-4053-6 |
| [24] | Fornasier, M., Banach frames for α-modulation spaces, Appl. Comput. Harmon. Anal., 22, 2, 157-175 (2007) · Zbl 1203.42039 · doi:10.1016/j.acha.2006.05.008 |
| [25] | Gröchenig, K., Describing functions: atomic decompositions versus frames, Monatsh. Math., 112, 3, 1-41 (1991) · Zbl 0736.42022 · doi:10.1007/BF01321715 |
| [26] | Gröchenig, K.; Piotrowski, M., Molecules in coorbit spaces and boundedness of operators, Studia Math., 192, 1, 61-77 (2009) · Zbl 1167.42007 · doi:10.4064/sm192-1-6 |
| [27] | Gröchenig, K.; Toft, J., Isomorphism properties of Toeplitz operators in time-frequency analysis, J. d’Analyse, 114, 1, 255-283 (2011) · Zbl 1243.42037 · doi:10.1007/s11854-011-0017-8 |
| [28] | K. Gröchenig, J. Toft, The range of localization operators and lifting theorems for modulation and Bargmann-Fock spaces. Trans. Am. Math. Soc. 23 (2012) · Zbl 1283.46021 |
| [29] | Heil, C.; Walnut, D. F., Continuous and discrete wavelet transforms, SIAM Rev., 31, 628-666 (1989) · Zbl 0683.42031 · doi:10.1137/1031129 |
| [30] | Hedenmalm, H.; Korenblum, B.; Zhu, K., Theory of Bergman Spaces (2000), New York: Springer, New York · Zbl 0955.32003 · doi:10.1007/978-1-4612-0497-8 |
| [31] | G. Kutyniok, et al. (eds.) Shearlets. Multiscale Analysis for Multivariate Data (Birkhäuser, Boston, MA). Appl. Numer. Harmonic Anal. 105-144 (2012) · Zbl 1237.42001 |
| [32] | Mantoiu, M., Quantization rules, Hilbert algebras and coorbit spaces for families of bounded operators I, Arxiv preprint, arXiv, 1203-6347 (2012) |
| [33] | Pap, M.; Schipp, F., The voice transform on the Blaschke group III, Publ. Math. Debrecen, 75, 1-2, 263-283 (2009) · Zbl 1212.43006 |
| [34] | Pap, M., The voice transform generated by a representation of the Blaschke group on the weighted Berman spaces, Ann. Univ. Sci. Budapest. Sect. Comp., 33, 321-342 (2010) · Zbl 1240.43006 |
| [35] | Pap, M., Properties of the voice transform of the Blaschke group and connections with atomic decomposition results in the weighted Bergman spaces, J. Math. Anal. Appl., 389, 1, 340-350 (2012) · Zbl 1235.43005 · doi:10.1016/j.jmaa.2011.11.060 |
| [36] | Perälä, A.; Taskinen, J.; Virtanen, J., New results and open problems on Toeplitz operators in Bergman spaces, New York J. Math., 17a, 147-164 (2011) · Zbl 1237.47035 |
| [37] | Rauhut, H., Banach frames in coorbit spaces consisting of elements which are invariant under symmetry groups, Appl. Comput. Harmon. Anal., 18, 1, 94-122 (2005) · Zbl 1087.46026 · doi:10.1016/j.acha.2004.09.002 |
| [38] | Rauhut, H., Coorbit space theory for quasi-Banach spaces, Studia Math., 180, 3, 237-253 (2007) · Zbl 1122.42018 · doi:10.4064/sm180-3-4 |
| [39] | Rauhut, H.; Ullrich, T., Generalized coorbit space theory and inhomogeneous function spaces of Besov-Lizorkin-Triebel type, J. Funct. Anal., 11, 3299-3362 (2011) · Zbl 1219.46035 · doi:10.1016/j.jfa.2010.12.006 |
| [40] | Romero, J. L., Characterization of coorbit spaces with phase-space covers, J. Funct. Anal., 262, 1, 59-93 (2012) · Zbl 1237.42015 · doi:10.1016/j.jfa.2011.09.005 |
| [41] | Schipp, F.; Wade, W., Transforms on Normed Fields, Leaflets in Mathematics (1995), Pecs: Janus Pannonius University, Pecs |
| [42] | Toft, J.; Boggiatto, P., Schatten classes for Toeplitz operators with Hilbert space windows on modulation spaces, Adv. Math., 217, 1, 305-333 (2008) · Zbl 1131.47026 · doi:10.1016/j.aim.2007.07.001 |
| [43] | K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, vol. 226 of Graduate Texts in Mathematics (Springer, New York, NY, 2005) · Zbl 1067.32005 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
