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Unified Theory: Fundamentals

  • Gianfranco Cariolaro

Abstract

The Unified Theory is based on two mathematical fundamentals: Abelian groups and the Haar integral. Abelian groups, developed in this chapter, will serve as the domains of signals, Fourier transforms and of their periodicity. The main problem is the identification of groups on where the Haar integral can be defined. Topology provides a clear answer to this problem: the groups must be locally compact Abelian (LCA). Another problem is the explicit formulation of these groups and of their operations. This is achieved through the technique of basis signature representation, where the basis identifies the geometry of the group and the signature specifies its nature (continuous, discrete or mixed).

The main topic of the chapter is the formulation and illustration of LCA groups, both one-dimensional and multidimensional. Another important topic developed is that of cells which are subsets of a group with specific properties and represent a fundamental tool for an advanced Signal Theory.

Keywords

Abelian Group Quotient Group Multiplicative Group Signal Classis Primitive Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    N. Bourbaki, General Topology, Parts 1 and 2 (Hermann, Paris, 1966) Google Scholar
  2. 2.
    G. Cariolaro, A unified signal theory (topological approach), in Proc. of Italy–USA Seminar on Digital Signal Processing, Portovenere, August 1981 Google Scholar
  3. 3.
    J.W.S. Cassels, An Introduction to the Geometry of Numbers (Springer, Berlin, 1997) MATHGoogle Scholar
  4. 4.
    E. Dubois, The sampling and reconstruction of time-varying imagery with application in video systems. Proc. IEEE 73, 502–523 (1985) CrossRefGoogle Scholar
  5. 5.
    E.P. Hansen, Theory of circular harmonic image reconstruction. J. Opt. Soc. Am. 71, 304–308 (1981) CrossRefGoogle Scholar
  6. 6.
    P. Halmos, Measure Theory (Van Nostrand, Princeton, 1950) MATHGoogle Scholar
  7. 7.
    R. Manducchi, G.M. Cortellazzo, G.A. Mian, Multistage sampling structure conversion of video signals. IEEE Trans. Signal Process. 41, 325–340 (1993) Google Scholar
  8. 8.
    W. Rudin, Fourier Analysis on Groups (Interscience, New York, 1962) MATHGoogle Scholar
  9. 9.
    A. Weil, L’Integration dans les Groupes Topologiques (Hermann, Paris, 1940) Google Scholar

Additional Bibliography on Topological Groups

  1. 10.
    N. Bourbaki, Integration (Hermann, Paris, 1959) Google Scholar
  2. 11.
    N. Bourbaki, Theory of Sets (Addison–Wesley/Hermann, Reading, 1968) MATHGoogle Scholar
  3. 12.
    A. Haar, Der Massbegriff in der Theorie der kontinuierlichen Gruppen. Ann. Math. 34, 147–169 (1933) MathSciNetCrossRefGoogle Scholar
  4. 13.
    E. Hewitt, K.A. Ross, Abstract Harmonic Analysis, vols. 1, 2 (Springer, Berlin, 1963) MATHCrossRefGoogle Scholar
  5. 14.
    P.J. Higgins, Introduction to Topological Groups (Cambridge University Press, London, 1974) MATHGoogle Scholar
  6. 15.
    L.H. Loomis, An Introduction to Abstract Harmonic Analysis (Van Nostrand, New York, 1953) MATHGoogle Scholar
  7. 16.
    L. Nachbin, The Haar Integral (Van Nostrand, Princeton, 1965) MATHGoogle Scholar
  8. 17.
    A. Orsatti, Introduzione ai Gruppi Abeliani Astratti e Topologici (Pitagora Ed., Bologna, 1979) MATHGoogle Scholar
  9. 18.
    L.S. Pontryagin, Topological Groups (Princeton University Press, Princeton, 1946) Google Scholar
  10. 19.
    M. Stroppel, Locally Compact Groups (European Mathematical Society, Zurich, 2006) MATHCrossRefGoogle Scholar

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© Springer-Verlag London Limited 2011

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