Abstract
In this chapter, we will provide a comprehensive overview of shearlet coorbit theory. We will present an almost self-contained introduction into coorbit theory which is the basis for all our investigations. We also discuss the group theoretical background of the continuous shearlet transform, and we explain how the shearlet transform can be combined with coorbit theory. By proceeding this way we can establish new smoothness spaces, the shearlet coorbit spaces. The structure of these spaces will be discussed in detail. In particular, we derive density, embedding, and trace results for the shearlet coorbit spaces.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)
Bergh, J., Löfström, J.: Interpolation Spaces: An Introduction. Springer, Berlin/ Heidelberg/New York (1976)
Borup, L., Nielsen, M.: Frame decomposition of decomposition spaces. J. Fourier Anal. Appl. 13(1), 39–70 (2007)
Candès, E.J., Donoho, D.L.: Ridgelets: A key to higher-dimensional intermittency? Phil. Trans. R. Soc. 357(1760), 2495–2509 (1999)
Candès, E.J., Donoho, D.L.: New tight frames of curvelets and optimal representations of objects with piecewise C 2 singularities. Comm. Pure Appl. Math. 57(2), 219–266 (2004)
Dahlke, S., Häuser, S., Steidl, G., Teschke, G.: Shearlet coorbit spaces: traces and embeddings in higher dimensions. Monatsh. Math. 169(1), 15–32 (2013)
Dahlke, S., Häuser, S., Teschke, G.: Coorbit space theory for the Toeplitz shearlet transform. Int. J. Wavelets Multiresolut. Inf. Process. 10(4) (2012)
Dahlke, S., Kutyniok, G., Maass, P., Sagiv, C., Stark, H.G., Teschke, G.: The uncertainty principle associated with the continuous shearlet transform. Int. J. Wavelets Multiresolut. Inf. Process. 6(2), 157–181 (2008)
Dahlke, S., Kutyniok, G., Steidl, G., Teschke, G.: Shearlet coorbit spaces and associated Banach frames. Appl. Comput. Harmon. Anal. 27(2), 195–214 (2009)
Dahlke, S., Steidl, G., Teschke, G.: Coorbit spaces and Banach frames on homogeneous spaces with applications to analyzing functions on spheres. Adv. Comput. Math. 21(1–2), 147–180 (2004)
Dahlke, S., Steidl, G., Teschke, G.: The continuous shearlet transform in arbitrary space dimensions. J. Fourier Anal. Appl. 16(3), 340–364 (2010)
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)
Dahlke, S., Teschke, G.: The continuous shearlet transform in higher dimensions: variations of a theme. In: Danellis, C.W. (ed.) Group Theory: Classes, Representations and Connections, and Applications, pp. 167–174. Nova Science, New York (2010)
Do, M.N., Vetterli, M.: Contourlets: a directional multiresolution image representation. In: Proceedings of the International Conference on Image Processing, vol. 1, pp. 357–360 (2002)
Feichtinger, H.G., Gröchenig, K.: A unified approach to atomic decompositions via integrable group representations. In: Function Spaces and Applications, (Lund, 1986), Lecture Notes in Mathematics, vol. 1302, pp. 52–73. Springer, Berlin/Heidelberg (1988)
Feichtinger, H.G., Gröchenig, K.: Banach spaces related to integrable group representations and their atomic decompositions, Part I. J. Funct. Anal. 86(2), 307–340 (1989)
Feichtinger, H.G., Gröchenig, K.: Banach spaces related to integrable group representations and their atomic decompositions. Part II. Monatsh. Math. 108(2), 129–148 (1989)
Feichtinger, H.G., Gröchenig, K.: Non-orthogonal wavelet and Gabor expansions and group representations. In: Ruskai, M.B., et al. (eds.) Wavelets and Their Applications, pp. 353–376. Jones and Bartlett, Boston (1992)
Fonseca, I., Leoni, G.: Modern Methods in the Calculus of Variations: L p Spaces. Springer, New York (2007)
Frazier, M., Jawerth, B.: Decomposition of Besov spaces. Indiana Univ. Math. J. 34(4), 777–799 (1985)
Gelfand, I.M., Vilenkin, N.Y.: Generalized Functions, Vol. 4. Applications of Harmonic Analysis. Academic Press, New York (1964)
Gröchenig, K.: Describing functions: atomic decompositions versus frames. Monatsh. Math. 112(1), 1–42 (1991)
Gröchenig, K.: Foundations of Time-Frequency Analysis. Birkhäuser, Boston (2001)
Gröchenig, K., Piotrowski, M.: Molecules in coorbit spaces and boundedness of operators. Stud. Math. 192(1), 61–77 (2009)
Guo, K., Kutyniok, G., Labate, D.: Sparse multidimensional representations using anisotropic dilation and shear operators. In: Chen, G., Lai, M.J. (eds.) Wavelets und Splines, pp. 189–201. Nashboro Press, Athens (2006)
Hedberg, L.I., Netrusov, Y.: An Axiomatic Approach to Function Spaces, Spectral Synthesis, and Luzin Approximation, 882 edn. American Mathematical Society, Providence (2007)
Jonsson, A., Wallin, H.: Function Spaces on Subsets of \(\mathbb{R}^{n}\). Harwood Academic, New York (1984)
Kittipoom, P., Kutyniok, G., Lim, W.Q.: Construction of compactly supported shearlet frames. Constr. Approx. 35(1), 21–72 (2012)
Kutyniok, G., Labate, D. (eds.): Shearlets: Multiscale Analysis for Multivariate Data. Birkhäuser, Boston (2012)
Kutyniok, G., Lemvig, J., Lim, W.Q.: Compactly supported shearlets. Proceedings of Approximation Theory XIII, pp. 163–186 (2012)
Schneider, C.: Besov spaces with positive smoothness. Dissertation, University of Leipzig (2009)
Triebel, H.: Theory of Function Spaces I. Birkhäuser, Boston (1983)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Dahlke, S., Häuser, S., Steidl, G., Teschke, G. (2015). Shearlet Coorbit Theory. In: Dahlke, S., De Mari, F., Grohs, P., Labate, D. (eds) Harmonic and Applied Analysis. Applied and Numerical Harmonic Analysis, vol 68. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-18863-8_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-18863-8_3
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-18862-1
Online ISBN: 978-3-319-18863-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)