Fractional transformations of generalised functions

Khaula Naeem Khan, Wilson Lamb, Adam C. McBride

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)
99 Downloads (Pure)


A distributional theory of fractional transformations is developed. A constructive approach, based on the eigenfunction expansion method pioneered by A. H. Zemanian, is used to produce an appropriate space of test functions and corresponding space of generalised functions. The fractional transformations that are defined are shown to form an equicontinuous group of operators on the space of test functions and a weak continuous group on the space of generalised functions. Integral representations for the fractional transformations are also obtained under certain conditions. The fractional Fourier transformation is considered as a particular case of our general theory.
Original languageEnglish
Pages (from-to)471-490
Number of pages20
JournalIntegral Transforms and Special Functions
Issue number6
Publication statusPublished - 13 May 2009


  • fractional integral transforms
  • semigroups of operators
  • generalised functions
  • mathematics
  • statistics


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