Abstract
In this article we construct orthonormal bases for bi-variate anisotropic α-modulation spaces. The construction is based on generating a nice anisotropic α-covering and using carefully selected tensor products of univariate brushlet functions with regards to this covering. As an application, we show that n-term nonlinear approximation with the orthonormal bases in certain anisotropic α-modulation spaces can be completely characterized.
Similar content being viewed by others
References
Auscher, P., Weiss, G., Wickerhauser, M.V.: Local sine and cosine bases of Coifman and Meyer and the construction of smooth wavelets. In: Wavelets, Wavelet Analysis and its Applications, vol. 2, pp. 237–256. Academic Press, Boston (1992)
Borup L., Nielsen M.: Approximation with brushlet systems. J. Approx. Theory 123(1), 25–51 (2003)
Borup L., Nielsen M.: Banach frames for multivariate α-modulation spaces. J. Math. Anal. Appl. 321(2), 880–895 (2006)
Borup L., Nielsen M.: Frame decomposition of decomposition spaces. J. Fourier Anal. Appl. 13(1), 39–70 (2007)
DeVore R.A., Jawerth B., Lucier B.J.: Image compression through wavelet transform coding. IEEE Trans. Inform. Theory 38(2, part 2), 719–746 (1992)
DeVore R.A., Jawerth B., Popov V.: Compression of wavelet decompositions. Am. J. Math. 114(4), 737–785 (1992)
Feichtinger H.G.: Banach spaces of distributions defined by decomposition methods. II. Math. Nachr. 132, 207–237 (1987)
Feichtinger H.G., Gröbner P.: Banach spaces of distributions defined by decomposition methods. I. Math. Nachr. 123, 97–120 (1985)
Garrigós G., Hernández E.: Sharp Jackson and Bernstein inequalities for N-term approximation in sequence spaces with applications. Indiana Univ. Math. J. 53(6), 1739–1762 (2004)
Gribonval R., Nielsen M.: Nonlinear approximation with dictionaries. I. Direct estimates. J. Fourier Anal. Appl. 10(1), 51–71 (2004)
Grobner, P.: Banachraeume glatter Funktionen und Zerlegungsmethoden. ProQuest LLC, Ann Arbor, Thesis (Dr. Natw.)–Technische Universitaet Wien (Austria) 1992
Hernández, E., Weiss, G.: A first course on wavelets. Studies in Advanced Mathematics. CRC Press, Boca Raton (With a foreword by Yves Meyer) (1996)
Kalton, N.: Quasi-Banach spaces. In: Handbook of the geometry of Banach spaces, vol. 2, pp. 1099–1130. North-Holland, Amsterdam (2003)
Kalton, N.J., Peck, N.T., Roberts, J.W.: An F-space sampler. In: London Mathematical Society Lecture Note Series, vol. 89. Cambridge University Press, Cambridge (1984)
Kyriazis G., Petrushev P.: New bases for Triebel-Lizorkin and Besov spaces. Trans. Am. Math. Soc. 354(2), 749–776 (2002) (electronic)
Laeng E.: Une base orthonormale de L 2(R) dont les éléments sont bien localisés dans l’espace de phase et leurs supports adaptés à toute partition symétrique de l’espace des fréquences. C. R. Acad. Sci. Paris Sér. I Math. 311(11), 677–680 (1990)
Meyer F.G., Coifman R.R.: Brushlets: a tool for directional image analysis and image compression. Appl. Comput. Harmon. Anal. 4(2), 147–187 (1997)
Meyer, Y.: Wavelets and operators. In: Cambridge Studies in Advanced Mathematics, vol. 37. Cambridge University Press, Cambridge (1992)
Nielsen M.: Orthonormal bases for α-modulationspaces. Collect. Math. 61(2), 173–190 (2010)
Rasmussen, K.N., Nielsen, M.: Compactly supported frames for decomposition spaces. J. Fourier Anal. Appl. (2011, in press)
Triebel, H.: Theory of function spaces. In: Monographs in Mathematics, vol. 78. Birkhäuser, Basel (1983)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rasmussen, K.N. Orthonormal bases for anisotropic α-modulation spaces. Collect. Math. 63, 109–121 (2012). https://doi.org/10.1007/s13348-011-0052-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13348-011-0052-x