EPFL
 Biomedical Imaging GroupSTI
EPFL
  Publications
English only   BIG > Publications > Characteristic Functionals


 CONTENTS
 Home Page
 News & Events
 People
 Publications
 Tutorials and Reviews
 Research
 Demos
 Download Algorithms

 DOWNLOAD
 PDF
 Postscript
 All BibTeX References

On the Continuity of Characteristic Functionals and Sparse Stochastic Modeling

J. Fageot, A. Amini, M. Unser

The Journal of Fourier Analysis and Applications, vol. 20, no. 6, pp. 1179-1211, December 2014.



The characteristic functional is the infinite-dimensional generalization of the Fourier transform for measures on function spaces. It characterizes the statistical law of the associated stochastic process in the same way as a characteristic function specifies the probability distribution of its corresponding random variable. Our goal in this work is to lay the foundations of the innovation model, a (possibly) non-Gaussian probabilistic model for sparse signals. This is achieved by using the characteristic functional to specify sparse stochastic processes that are defined as linear transformations of general continuous-domain white Lévy noises (also called innovation processes). We prove the existence of a broad class of sparse processes by using the Minlos-Bochner theorem. This requires a careful study of the regularity properties, especially the Lp-boundedness, of the characteristic functional of the innovations. We are especially interested in the functionals that are only defined for p < 1 since they appear to be associated with the sparser kind of processes. Finally, we apply our main theorem of existence to two specific subclasses of processes with specific invariance properties.


@ARTICLE(http://bigwww.epfl.ch/publications/fageot1401.html,
AUTHOR="Fageot, J. and Amini, A. and Unser, M.",
TITLE="On the Continuity of Characteristic Functionals and Sparse
        Stochastic Modeling",
JOURNAL="The Journal of {F}ourier Analysis and Applications",
YEAR="2014",
volume="20",
number="6",
pages="1179--1211",
month="December",
note="")

© 2014 Birkhäuser. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from Birkhäuser.
This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.