Abstract
The theory of frames and non-orthogonal series expansions with respect to coherent states is extended to a general class of spaces, the so-called coorbit spaces. Special cases include wavelet expansions for the Besov-Triebel-Lizorkin spaces, Gabortype expansions for modulation spaces, and sampling theorems for wavelet and Gabor transforms.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[B]Bargmann, V.: On a Hilbert space of analytic functions and an associated integral transform. Part I. Comm. Pure Appl. Math.14, 187–214 (1961).
[BA1]Bastiaans, M. J.: Signal description by means of local frequency spectrum. SPIE373, 49–62 (1981).
[BA]Battle, G.: Heisenberg proof of the Balian-Low theorem. Lett. Math. Phys.15, 175–177 (1988).
[BO]Bohnké, G.: Treillis d'ondellettes associés aux groupes de Lorentz. (Preprint.)
[CMS]Coifman, R., Meyer, Y., Stein, E.: Some new function spaces and their applications to harmonic analysis. J. Funct. Anal.62, 304–335 (1985).
[CR]Coifman, R. Rochberg, R.: Representation theorems for holomorphic and harmonic functions. Astérisque77, 11–65 (1980).
[D1]Daubechies, I. The wavelet transform, time-frequency localization and signal analysis. IEEE Trans. Inform. Theory36, 961–1005 (1990).
[D2]Daubechies, I.: Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math.41, 909–996 (1988).
[DG]Daubechies, I., Grossmann, A.: Frames in the Bargmann space of entire functions. Comm. Pure Appl. Math.16, 151–169 (1988).
[DGM]Daubechies, I., Grossmann, A., Meyer, Y.: Painless nonorthogonal expansions. J. Math. Phys.27, 1271–1283 (1986).
[DS]Duffin, R., Schaeffer, A.: A class of nonharmonic Fourier series Trans. Amer. Math. Soc.72, 341–366 (1952).
[DM]Duflo, M., Moore, C.C.: On the regular representation of a non-unimodular locally compact group. J. Funct. Anal.21, 209–243 (1976).
[F1]Feichtinger, H. G.: Banach convolution algebras of Wiener's type. In: Functions, Series, Operators. 509–524. Proc. Conf., Budapest 1980. Amsterdam: North Holland. 1983.
[F2]Feichtinger, H. G.: Atomic characterizations of modulation spaces through Gabor type representations. Rocky Mountain J. Math.19, 113–126 (1989).
[F3]Feichtinger, H. G.: Modulation spaces on locally compact abelian groups. Techn. Report. University of Vienna, 1983.
[FG1]Feichtinger, H. G., Gröchenig, K.: A unified approach to atomic decompositions via integrable group representations. In: Proc. Conf. “Functions Spaces and Applications”. (M. Cwikel et al. eds.) pp. 52–73. Lect. Notes Math. 1302, Berlin-Heidelberg-New York: Springer. 1988.
[FG2]Feichtinger, H. G., Gröchenig, K.: Banach Spaces related to integrable group representations and their atomic decompositions I. J. Funct. Anal.86, 307–340 (1989).
[FG3]Feichtinger, H. G., Gröchenig, K.: Banach Spaces related to integrable group representations and their atomic decompositions II. Mh. Math.108, 129–148 (1989).
[FJ1]Frazier, M., Jawerth, B.: Decompositon of Besov spaces. Indiana Univ. Math. J.34, 777–799 (1985).
[FJ2]Frazier, M., Jawerth, B.: The φ-transform and decompositions of distribution spaces. In: Proc. Conf. “Functions Spaces and Applications” (Cwikel, M., et al., eds.) Lect. Notes Math. 1302. Berlin-Heidelberg-New York: Springer. 1988.
[FJ3]Frazier, M., Jawerth, B.: A discrete transform and decompositions of distribution spaces. J. Funct. Anal.93, 34–170 (1990).
[FS]Folland, G., Stein, E.: Hardy Spaces on Homogeneous Groups. Princeton: Univ. Press. 1982.
[G1]Gröchenig, K.: Analyses multi-échelles et bases d'ondelettes. C. R. Acad. Sci. Paris305, 13–15 (1987).
[G2]Gröchenig, K.: Unconditional bases in translation and dilation invariant function spaces on ℝn. In: Constructive Theory of Functions. Proc. Conf. Varna 1987 (Sendov, B. et al., eds.), pp. 174–183. Bulgarian Acad. Sci. 1988.
[GMP]Grossmann, A., Morlet, J., Paul, T.: Transforms associated to square integrable group representations I. J. Math. Phys.26, 2473–2479 (1985).
[HW]Heil, C. E., Walnut, D. F.: Continuous and discrete wavelet transforms. SIAM Review31, 628–666 (1989).
[JPR]Janson, S., Peetre, J., Rochberg, R.: Hankel forms and the Fock space. Revista Math. Iberoam.3, 61–138 (1987).
[KS]Klauder, J. R., Skagerstam, B. S.: Coherent States. Singapore: World Scientific. 1985.
[LM]Lemarie, P. G., Meyer, Y.: Ondelettes et bases hilbertiennes. Revista Math. Iberoam.2, 1–18 (1986).
[L]Luecking, D.: Representation and duality in weighted spaces of analytic functions. Indiana Univ. Math. J.34, 319–336 (1985).
[M]Meyer, Y. Ondelettes, Vol.I. Paris: Hermann. 1990.
[MA]Mallat, S.: Multiresolution approximation and wavelet bases ofL 2. Trans. Amer. Math. Soc.315, 69–87 (1989).
[RT]Ricci, F., Taibleson, M.: Boundary values of harmonic functions in mixed norm spaces and their atomic structure. Ann. Scuola Norm. Sup. Pisa, Ser. IV,10, 1–54 (1983).
[R]Rochberg, R.: Decomposition theorems for Bergman spaces and their applications. In: Operators and Function Theory. (Powers, S. C., ed.) Reidel. 1985.
[SV]Sharpley, R., De Vore, R.: Maximal Functions Measuring Smoothness. Memoirs Amer. Math. Soc.293, (1984).
[S]Stettinger, F.: Banachräume von Funktionen und Oszillation. Ph.D. Thesis. Univ. of Vienna. 1983.
[T]Triebel, H.: Characterizations of Besov-Hardy-Sobolev-spaces: A unified approach. J. Approx. Theory52, 162–203 (1988).
[T2]Triebel, H.: Theory of function Spaces. Leipzig: Akad. Verlagsges. 1983.
[Y]Young, R. M.: An Introduction to Nonharmonic Fourier Series. New York: Academic Press. 1980.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Gröchenig, K. Describing functions: Atomic decompositions versus frames. Monatshefte für Mathematik 112, 1–42 (1991). https://doi.org/10.1007/BF01321715
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01321715