Construction and Properties of Canonical Transforms

Abstract

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.