Integral Equations and Operator Theory

, Volume 68, Issue 2, pp 163–191 | Cite as

Discrete-Time Multi-Scale Systems

Article

Abstract

We introduce multi-scale filtering by the way of certain double convolution systems. We prove stability theorems for these systems and make connections with function theory in the polydisc. Finally, we compare the framework developed here with the white noise space framework, within which a similar class of double convolution systems has been defined earlier.

Mathematics Subject Classification (2000)

Primary 94A12 47N70 46E22 Secondary 93D25 42A70 

Keywords

Discrete-scale transformation scale invariance linear systems self-similarity reproducing kernels 

References

  1. 1.
    Alpay, D.: The Schur algorithm, reproducing kernel spaces and system theory. In: SMF/AMS Texts and Monographs, vol. 5, American Mathematical Society, Providence, RI (2001) Translated from the 1998 French original by Stephen S. WilsonGoogle Scholar
  2. 2.
    Alpay D., Dym H.: Hilbert spaces of analytic functions, inverse scattering and operator models, I. Integr. Equ. Oper. Theory 7, 589–641 (1984)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Alpay D., Levanony D.: Rational functions associated to the white noise space and related topics. Potential Anal. 29, 195–220 (2008)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Alpay, D., Levanony, D.: Linear stochastic systems: a white noise approach. Acta Appl. Math. (to appear)Google Scholar
  5. 5.
    Alpay, D., Levanony, D., Pinhas, A.: Linear State space theory in the white noise space setting (Preprint)Google Scholar
  6. 6.
    Alpay, D., Levanony, D., Mboup, M.: Double convolution systems (in preparation)Google Scholar
  7. 7.
    Alpay D., Mboup M.: A characterization of Schur multipliers between character-automorphic Hardy spaces. Integ. Equ. Oper. Theory 62, 455–463 (2008)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Alpay, D., Mboup, M.: Transformée en échelle de signaux stationnaires. Comptes-Rendus mathématiques (Paris), vol. 347, Issues 11–12, June 2009, pp. 603–608Google Scholar
  9. 9.
    Alpay, D., Mboup, M.: A natural transfer function space for linear discrete time-invariant and scale-invariant systems. In: Proceedings of NDS09, Thessaloniki, Greece, June 29–July 1, 2009Google Scholar
  10. 10.
    Aronszajn N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 227–404 (1950)MathSciNetGoogle Scholar
  11. 11.
    Ball, J., Bolotnikov, V.: Boundary interpolation for contractive-valued functions on circular domains in \({\mathbb{C}^{n}}\). In: Current Trends in Operator Theory and its Applications, Oper. Theory Adv. Appl., vol. 149, pp. 107–132. Birkhäuser, Basel (2004)Google Scholar
  12. 12.
    Ball, J., Trent, T., Vinnikov, V.: Interpolation and commutant lifting for multipliers on reproducing kernel Hilbert spaces. In: Proceedings of Conference in Honor of the 60–th Birthday of M.A. Kaashoek, Operator Theory: Advances and Applications, vol. 122, pp. 89–138. Birkhauser (2001)Google Scholar
  13. 13.
    Ball, J., Vinnikov, V.: Functional models for representations of the Cuntz algebra. In: Operator Theory, Systems Theory and Scattering Theory: Multidimensional Generalizations, Oper. Theory Adv. Appl., vol. 157, pp. 1–60. Birkhäuser, Basel (2005)Google Scholar
  14. 14.
    de Branges L., Shulman L.A.: Perturbation theory of unitary operators. J. Math. Anal. Appl. 23, 294–326 (1968)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Deitmar, A.: A first course in harmonic analysis. Universitext, 2nd edn. Springer (2005)Google Scholar
  16. 16.
    Ford L.R.: Automorphic functions, 2nd edn. Chelsea, New-York (1915)MATHGoogle Scholar
  17. 17.
    Freitag, E., Busam, R.: Complex Analysis. Springer (2005)Google Scholar
  18. 18.
    Guelfand, I.M., Graev, M.I., Vilenkin, N.Ja.: Les distributions. Tome 5. Géométrie intégrale et théorie des représentations. Dunod, Paris (1970)Google Scholar
  19. 19.
    Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis, vol. I/II, Springer, Berlin, Göttingen Heidelberg (1963/1970)Google Scholar
  20. 20.
    Hida T., Kuo H., Potthoff J., Streit L.: White noise. An infinite-dimensional calculus. Mathematics and its Applications, vol. 253. Kluwer, Dordrecht (1993)Google Scholar
  21. 21.
    Hida T.: White noise analysis: part I. Theory in progress. Taiwan. J. Math. 7, 541–556 (2003)MATHMathSciNetGoogle Scholar
  22. 22.
    Holden, H., Øksendal, B., Ubøe, J., Zhang, T.: Stochastic partial differential equations. In: Probability and its Applications. Birkhäuser Boston Inc., Boston, MA (1996)Google Scholar
  23. 23.
    Katok, S.: Fuchsian groups. Chicago Lecture Notes in Mathematics, University of Chicago Press (1992)Google Scholar
  24. 24.
    Kreĭn, M.G., Nudelman, A.A.: The Markov moment problem and extremal problems. In: Translations of Mathematical Monographs, vol. 50, American Mathematical Society, Providence, RI (1977)Google Scholar
  25. 25.
    Leland W., Taqqu M., Willinger W., Wilson D.: On the self-similar nature of Ethernet traffic (extended version). IEEE/ACM Trans. Netw. 2(1), 1–15 (1994)CrossRefGoogle Scholar
  26. 26.
    Mallat, S.: Une exploration des signaux en ondelettes. Les éditions de l’École Polytechnique (2000)Google Scholar
  27. 27.
    Mandelbrot B.B., Van Ness W.: Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10(4), 422–437 (1968)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Mboup, M.: A character-automorphic Hardy spaces approach to discrete-time scale-invariant systems. In: Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems, Kyoto, Japan, July 24–28, 2006, pp. 183–188 (2006)Google Scholar
  29. 29.
    Putinar M.: Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42(3), 969–984 (1993)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Saitoh, S.: Theory of reproducing kernels and its applications. In: Longman scientific and technical, vol. 189 (1988)Google Scholar
  31. 31.
    Yuditskii, P.: Two remarks on Fuchsian groups of Widom type. In: Operator Theory: Advances and Applications, vol. 123, pp. 527–537. Birkhauser (2001)Google Scholar

Copyright information

© Birkhäuser / Springer Basel AG 2010

Authors and Affiliations

  1. 1.Department of mathematicsBen-Gurion University of the NegevBeershebaIsrael
  2. 2.CReSTIC-UFR des Sciences Exactes et NaturellesUniversité de Reims Champagne-ArdenneReims Cedex 2France

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