Chromatic Expansions and the Bargmann Transform

  • Ahmed I. ZayedEmail author


Chromatic series expansions of bandlimited functions have recently been introduced in signal processing with promising results. Chromatic series share similar properties with Taylor series insofar as the coefficients of the expansions, which are called chromatic derivatives, are based on the ordinary derivatives of the function, but unlike Taylor series, chromatic series have more practical applications.The Bargmann transform was introduced in 1961 by V. Bargmann who showed, among other things, that the Bargmann transform is a unitary transformation from L 2(I​​R n ) onto the Bargmann–Segal–Foch space \(\mathfrak{F}\) on which Foch’s operator solutions to some equations in quantum mechanics are realized.The goal of this article is to survey results on chromatic derivatives and explore the connection between chromatic derivatives and series on the one hand and the Bargmann transform and the Bargmann–Segal–Foch space on the other hand.


Bargmann Transform Chromene Derivatives Serum Chromium Band-limited Functions Normalized Hermite Functions 
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.


  1. 1.
    Bargmann V, Butera P, Girardello L, Klauder JR (1971) On the completeness of the coherent states. Rep Math Phys 2:221–228MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bargmann V (1967) On a Hilbert space of analytic functions and an associated integral transform, Part II, A family of related function spaces application to distribution theory. Comm Pure Appl Math 20:1–101MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bargmann V (1961) On a Hilbert space of analytic functions and an associated integral transform, Part I. Comm Pure Appl Math 14:187–214MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Cushman M, Herron T (2001) The general theory of chromatic derivatives. Kromos technology technical report, Los AltosGoogle Scholar
  5. 5.
    Dunkl C, Xu Y (2001) Orthogonal polynomials of several variables. Encyclopedia of mathematics, Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  6. 6.
    Herron T, Byrnes J (2001) Families of orthogonal differential operators for signal processing. Kromos technology technical report, Los AltosGoogle Scholar
  7. 7.
    Foch V (1928) Verallgemeinerung und Lösung der Diracschen statistischen Gleichung. Z Phys 49:339–357CrossRefGoogle Scholar
  8. 8.
    Gradshteyn I, Ryzhik I (1965) Tables of integrals, series, and products. Academic, New YorkGoogle Scholar
  9. 9.
    Ignjatovic A, Zayed A (2011) Multidimensional chromatic derivatives and series expansions. Proc Amer Math Soc 139(10):3513–3525MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Ignjatovic A (2009) Chromatic derivatives, chromatic expansions and associated function spaces. East J Approx 15(2):263–302MathSciNetGoogle Scholar
  11. 11.
    Ignjatovic A (2007) Local approximations based on orthogonal differential operators. J Fourier Anal & Applns 13:309–330MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Ignjatovic A (2001) Numerical differentiation and signal processing. Proc International Conference Information, Communications and Signal Processing (ICICS), SingaporeGoogle Scholar
  13. 13.
    Ignjatovic A (2001) Local approximations and signal processing. Kromos technology technical report, Los AltosGoogle Scholar
  14. 14.
    Ignjatovic A (2001) Numerical differentiation and signal processing. Kromos technology technical report, Los AltosGoogle Scholar
  15. 15.
    Ignjatovic A, Carlin N (1999) Signal processing with local behavior, Provisional Patent Application, (60)-143,074, US Patent Office, Patent issued as US 6313778, June 2001Google Scholar
  16. 16.
    Janssen AJ (1982) Bargmann transform, zak transform, and coherent states. J Math Phys 23:720–731MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Janssen AJEM 1981) Gabor representation of generalized functions. J Math Anal Appl 83: 377–394MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Klauder J, Skagerstam BS (1985) Coherent states. World scientific, SingaporezbMATHGoogle Scholar
  19. 19.
    Klauder J, Sudarshan EC (1968) Fundamentals of quantum optics. W. A. Benjamin, New YorkGoogle Scholar
  20. 20.
    Marchenko V (1986) Sturm-Liouville operators and their applications. Birkhauser, New YorkGoogle Scholar
  21. 21.
    Naimark MA (1967) Linear differential operators I. George Harrap, LondonzbMATHGoogle Scholar
  22. 22.
    Narasimha M, Ignjatovic A, Vaidyanathan P (2002) Chromatic derivative filter banks. IEEE Signal Processing Lett 9(7):215–216CrossRefGoogle Scholar
  23. 23.
    Szegö G (1975) Orthogonal polynomials. Amer Math Soc, Providence, RIzbMATHGoogle Scholar
  24. 24.
    Titchmarsh E (1962) Eigenfunction expansion I. Oxford University Press, LondonGoogle Scholar
  25. 25.
    Vaidyanathan P, Ignjatovic A, Narasimha M (2001) New sampling expansions of bandlimited signals based on chromatic derivatives. Proc 35th Asilomar Conf Signals, Systems and Computers, Monterey, pp 558–562Google Scholar
  26. 26.
    Walter G, Shen X (2005) A sampling expansion for non-bandlimited signals in chromatic derivatives. IEEE Trans Signal Process 53: 1291–1298MathSciNetCrossRefGoogle Scholar
  27. 27.
    Zayed AI (2011) Chromatic expansions of generalized functions. Integral Transforms and Special Functions 22(4–5):383–390MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Zayed AI (2010) Generalizations of chromatic derivatives and series expansions. IEEE Trans Signal Process 58(3):1638–1647MathSciNetCrossRefGoogle Scholar
  29. 29.
    Zayed AI (1993) Advances in Shannon’s sampling theory. CRC Press, Boca RatonzbMATHGoogle Scholar

Copyright information

© Springer New York 2013

Authors and Affiliations

  1. 1.Department of Mathematical SciencesDePaul UniversityChicagoUSA

Personalised recommendations