Skip to main content

Construction and Properties of Canonical Transforms

  • Chapter
Integral Transforms in Science and Engineering

Part of the book series: Mathematical Concepts and Methods in Science and Engineering ((MCSENG,volume 11))


In this chapter we present a class of integral transforms which we shall call canonical transforms. These constitute a parametrized continuum of transforms which include the Fourier, Laplace, Gauss-Weierstrass, and Bargmann transforms as particular cases. As these have arisen quite recently, we shall include brief historical sketches of the developments which led to their recognition and refer the interested reader to the research literature for a more rigorous treatment. Section 9.1 deals with real linear canonical transforms, while Section 9.2 enlarges the set to complex ones. The former appeared a couple of times before Moshinsky and Quesne (1974) called attention to their significance in connection with canonical transformations in quantum mechanics. A particular case of the latter was developed by Segal (1963) and Bargmann (1961) in order to formalize Fok’s boson calculus (1928). Section 9.3 shows that canonical transforms have a hyperdifferential operator realization in addition to the usual integral form. Several examples and exercises show the economy of concepts and computation introduced by this new technique.

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

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

USD 9.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight 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


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Author information

Authors and Affiliations


Rights and permissions

Reprints and permissions

Copyright information

© 1979 Springer Science+Business Media New York

About this chapter

Cite this chapter

Wolf, K.B. (1979). Construction and Properties of Canonical Transforms. In: Integral Transforms in Science and Engineering. Mathematical Concepts and Methods in Science and Engineering, vol 11. Springer, Boston, MA.

Download citation

  • DOI:

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4757-0874-5

  • Online ISBN: 978-1-4757-0872-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics