The integral transforms most often encountered in applications, notably the Fourier transform, owe their importance to the manner in which they map operations on the functions subject to the transformation onto operations on the transformed functions: the Fourier transform turns the operations of differentiation and of translation into operations of multiplication; and these two facts are closely connected in that differentiation is the infinitesimal operator of the group of translations of functions. There are numerous other familiar examples. Plancherel [4] and Kober [2], [3] have used particular functional equations in order to investigate properties of the functions which satisfy them. These considerations suggest that it will be useful to study the types of functional equations which arise in connection with linear transformations of function spaces on which a group of transformations acts.


Functional Equation Function Space Linear Transformation Compact Support Complete Solution 
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  1. [1]
    J. L. B. Cooper, Functional equations for linear transformations. Proc. London Math. Soc. (to appear).Google Scholar
  2. [2]
    H. Kober, On certain linear operations and relations between them. Proc. London Math. Soc. (3) 11 (1961), 434–456.CrossRefGoogle Scholar
  3. [3]
    H. Kober, On functional equations and bounded linear transformations. Proc. London Math. Soc. (3) 14 (1964), 495–519.CrossRefGoogle Scholar
  4. [4]
    M. Plancherel, Quelques remarques à propos d’une note de G. H. Hardy; the resultant of two Fourier kernels. Proc. Cambridge Philos. Soc. 33 (1937), 413–418.CrossRefGoogle Scholar

Copyright information

© Springer Basel AG 1969

Authors and Affiliations

  • J. L. B. Cooper
    • 1
  1. 1.Chelsea CollegeUniversity of LondonUK

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