Skip to main content
SpringerLink
Log in

Chromatic Expansions and the Bargmann Transform

  • Chapter
  • First Online:
Multiscale Signal Analysis and Modeling

  • 1542 Accesses

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    If x, ω ∈ I​​R d are both real vectors, then we denote their scalar product by x. ω. If at least one of u, z is complex, we denote their scalar product by ⟨u, z⟩.

References

  1. Bargmann V, Butera P, Girardello L, Klauder JR (1971) On the completeness of the coherent states. Rep Math Phys 2:221–228

    Article  MathSciNet  Google Scholar 

  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 20:1–101

    Article  MathSciNet  MATH  Google Scholar 

  3. Bargmann V (1961) On a Hilbert space of analytic functions and an associated integral transform, Part I. Comm Pure Appl Math 14:187–214

    Article  MathSciNet  MATH  Google Scholar 

  4. Cushman M, Herron T (2001) The general theory of chromatic derivatives. Kromos technology technical report, Los Altos

    Google Scholar 

  5. Dunkl C, Xu Y (2001) Orthogonal polynomials of several variables. Encyclopedia of mathematics, Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  6. Herron T, Byrnes J (2001) Families of orthogonal differential operators for signal processing. Kromos technology technical report, Los Altos

    Google Scholar 

  7. Foch V (1928) Verallgemeinerung und Lösung der Diracschen statistischen Gleichung. Z Phys 49:339–357

    Article  Google Scholar 

  8. Gradshteyn I, Ryzhik I (1965) Tables of integrals, series, and products. Academic, New York

    Google Scholar 

  9. Ignjatovic A, Zayed A (2011) Multidimensional chromatic derivatives and series expansions. Proc Amer Math Soc 139(10):3513–3525

    Article  MathSciNet  MATH  Google Scholar 

  10. Ignjatovic A (2009) Chromatic derivatives, chromatic expansions and associated function spaces. East J Approx 15(2):263–302

    MathSciNet  Google Scholar 

  11. Ignjatovic A (2007) Local approximations based on orthogonal differential operators. J Fourier Anal & Applns 13:309–330

    Article  MathSciNet  MATH  Google Scholar 

  12. Ignjatovic A (2001) Numerical differentiation and signal processing. Proc International Conference Information, Communications and Signal Processing (ICICS), Singapore

    Google Scholar 

  13. Ignjatovic A (2001) Local approximations and signal processing. Kromos technology technical report, Los Altos

    Google Scholar 

  14. Ignjatovic A (2001) Numerical differentiation and signal processing. Kromos technology technical report, Los Altos

    Google Scholar 

  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 2001

    Google Scholar 

  16. Janssen AJ (1982) Bargmann transform, zak transform, and coherent states. J Math Phys 23:720–731

    Article  MathSciNet  MATH  Google Scholar 

  17. Janssen AJEM 1981) Gabor representation of generalized functions. J Math Anal Appl 83: 377–394

    Article  MathSciNet  MATH  Google Scholar 

  18. Klauder J, Skagerstam BS (1985) Coherent states. World scientific, Singapore

    MATH  Google Scholar 

  19. Klauder J, Sudarshan EC (1968) Fundamentals of quantum optics. W. A. Benjamin, New York

    Google Scholar 

  20. Marchenko V (1986) Sturm-Liouville operators and their applications. Birkhauser, New York

    Google Scholar 

  21. Naimark MA (1967) Linear differential operators I. George Harrap, London

    MATH  Google Scholar 

  22. Narasimha M, Ignjatovic A, Vaidyanathan P (2002) Chromatic derivative filter banks. IEEE Signal Processing Lett 9(7):215–216

    Article  Google Scholar 

  23. Szegö G (1975) Orthogonal polynomials. Amer Math Soc, Providence, RI

    MATH  Google Scholar 

  24. Titchmarsh E (1962) Eigenfunction expansion I. Oxford University Press, London

    Google Scholar 

  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 558–562

    Google Scholar 

  26. Walter G, Shen X (2005) A sampling expansion for non-bandlimited signals in chromatic derivatives. IEEE Trans Signal Process 53: 1291–1298

    Article  MathSciNet  Google Scholar 

  27. Zayed AI (2011) Chromatic expansions of generalized functions. Integral Transforms and Special Functions 22(4–5):383–390

    Article  MathSciNet  MATH  Google Scholar 

  28. Zayed AI (2010) Generalizations of chromatic derivatives and series expansions. IEEE Trans Signal Process 58(3):1638–1647

    Article  MathSciNet  Google Scholar 

  29. Zayed AI (1993) Advances in Shannon’s sampling theory. CRC Press, Boca Raton

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, DePaul University, Chicago, IL, 60614, USA

    Ahmed I. Zayed

Authors
  1. Ahmed I. Zayed
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ahmed I. Zayed .

Editor information

Editors and Affiliations

  1. , Department of Mathematics, Ohio University, Ohio University 1, Athens, 45701, Ohio, USA

    Xiaoping Shen

  2. , Department of Mathematical Sciences, DePaul University, North Kenmore Avenue 2320, Chicago, 60614, Illinois, USA

    Ahmed I. Zayed

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer New York

About this chapter

Cite this chapter

Zayed, A.I. (2013). Chromatic Expansions and the Bargmann Transform. In: Shen, X., Zayed, A. (eds) Multiscale Signal Analysis and Modeling. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4145-8_6

Download citation

  • DOI: https://doi.org/10.1007/978-1-4614-4145-8_6

  • Published:

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4614-4144-1

  • Online ISBN: 978-1-4614-4145-8

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation