Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-26T10:51:29.805Z Has data issue: false hasContentIssue false
  • English
  • Français

Localization Operators for Ridgelet Transforms

Published online by Cambridge University Press:  17 July 2014

J. Li  and
M. W. Wong
J. Li
Affiliation:
Department of Mathematics and Statistics, York University 4700 Keele Street, Toronto, Ontario M3J 1P3, Canada
M. W. Wong*
Affiliation:
Department of Mathematics and Statistics, York University 4700 Keele Street, Toronto, Ontario M3J 1P3, Canada
*
⋆Corresponding author. E-mail: mwwong@mathstat.yorku.ca This research has been supported by the NaturalSciences and Engineering Research Council of Canada.
Get access

Abstract

We prove that localization operators associated to ridgelet transforms withLp symbols are boundedlinear operators on L2(Rn).Operators closely related to these localization operators are shown to be in the traceclass and a trace formula for them is given.

Keywords

Gabor transformswavelet transformscurvelet transformswavelet multipliersridgelet transformsRadon transformsresolution of the identity formulascontinuous inversion formulaslocalization operatorsL2-boundednesstrace class operatorstracesLidskii’s formula
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 5: Spectral problems , 2014 , pp. 194 - 203
Copyright
© EDP Sciences, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

E. J. Candès. Ridgelets: Theory and Applications. Ph.D. Thesis, Department of Statistics, Stanford University, 1998.
Candès, E. J., Donoho, D. L.. Ridgelets: a key to higher-dimensional intermittency? R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 357 (1999), 2495–2509. CrossRefGoogle Scholar
Candès, E. J., Donoho, D. L.. Continuous curvelet transform II. discretization and frames. Appl. Comput. Harmon. Anal., 19 (2005), 198–222. CrossRefGoogle Scholar
V. Catan\hbox{${\rm \breve{a}}$}˘a. Products of two-wavelet multipliers and their traces. in Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations, Operator Theory: Advances and Applications 205, Birkhäuser, 2010, 195–211.
Daubechies, I.. Time-frequency localization operators, a geometric phase space approach. IEEE Trans. Inform. Theory, 34 (1988), 605–612. CrossRefGoogle Scholar
I. Daubechies. Ten Lectures on Wavelets, SIAM, 1992.
Du, J., Wong, M. W.. Traces of wavelet multiplers. C. R. Math. Rep. Acad. Sci. Canada, 23 (2001), 148–152. Google Scholar
Goupillaud, P., Grossmann, A., Morlet, J.. Cycle-octave and related transforms in seismic signal analysis. Geoexploration, 23 (1984), 85–102. CrossRefGoogle Scholar
He, Z., Wong, M. W.. Wave multipliers and signals. J. Austral. Math. Soc. Ser. B, 40 (1999), 437–446. CrossRefGoogle Scholar
Landau, H. J., Pollak, H. O.. Prolate spheroidal wave functions, Fourier analysis and uncertainty. Bell Syst. Tech. J., 40 (1961), 65–84. CrossRefGoogle Scholar
Li, J., Wong, M. W.. Localization operators for curvelet transforms. J. Pseudo-Differ. Oper. Appl., 3 (2012), 121–143. CrossRefGoogle Scholar
Lidskii, V. B.. Nonself-adjoint operators with trace. Dokl. Akad. Nauk SSR, 125 (1959), 485–487; Amer. Math. Soc. Translations, 47 (1961), 43–46. Google Scholar
Pollak, H. J.. Prolate spheroidal wave functions, Fourier analysis and uncertainty, III. Bell Syst. Tech. J., 41 (1962), 1295–1336. Google Scholar
Slepian, D.. On bandwidth. Proc IEEE, 64 (1976), 292–300. CrossRefGoogle Scholar
Slepian, D.. Some comments on Fourier analysis, uncertainty and modeling. SIAM Rev., 25 (1983), 379–393. CrossRefGoogle Scholar
Slepian, D., Pollak, H. O.. Prolate spheroidal wave functions. Fourier analysis and uncertainty, I, Bell Syst. Tech. J., 40 (1961), 43–64. CrossRefGoogle Scholar
E. M. Stein, R. Shakarchi. Fourier Analysis: An Introduction. Princeton University Press, 2003.
M. W. Wong. Weyl Transforms. Springer, 1998.
M. W. Wong. Wavelet Transforms and Localization Operators. Birkhäuser, 2002.
M. W. Wong. Discrete Fourier Analysis. Birkhäuser, 2011.
Wong, M. W., Zhang, Z.. Traces of two-wavelet multipliers. Integr. Equ. Oper. Theory, 42 (2002), 498–503. CrossRefGoogle Scholar
Wong, M. W., Zhang, Z.. Trace class norm inequalities for wavelet multipliers. Bull. London Math. Soc., 34 (2002), 739–744. CrossRefGoogle Scholar