Remarks on isometric mappings between the Hardy–Szegö and Bergman–Selberg spaces: wavelet transforms and Laguerre functions

Abstract

It is observed that a certain isometric mapping between Hardy–Szegö and Bergman–Selberg spaces, studied by D-W Byun, generalizes a transformation arising from a family of special wavelet transforms studied by Grossman et al. in one of the early papers on wavelet theory. Furthermore, we will see that, when mapped under such transforms, the Fourier transforms of Laguerre functions play the same canonical role as do the Hermite functions when mapped via the Bargmann–Fock transform. As a byproduct we obtain an integral representation, with respect to area measure in the upper half plane, of the Fourier transforms of the Laguerre functions.

Keywords:

• Laguerre functions
• wavelets
• Bergman spaces
• Hardy spaces

1991 Mathematics Subject Classifications :

• 42C40
• 30H05
• 33C45

Acknowledgements

Partial financial assistance by the FCT grant SFRH/BPD/26078/2005 and the CMUC.

Additional information

Notes on contributors

Luís Daniel Abreu

Email: [email protected]