Abstract
The classical Hecke identity gives the Fourier transform of the product of a homogeneous harmonic polynomial h times the Gaussian e−1/2<...>. A similar formula is valid when the Gaussian is replaced by the tempered distribution ei/2<...>. It is shown that there is a similar identity when the inner product is replaced by an indefinite quadratic formq and h is a Л-harmonic distribution, where Л is the differential operator canonically associated toq. Another generalization is obtained in the context of representations of Jordan algebras, in the spirit of Herz's previous work on matrix spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Faraut, J. and Korányi, A. (1994).Analysis on Symmetric Cones, Oxford Mathematical Monographs, Clarendon Press, Oxford.
Gelfand, I.M. and Shilov, G. (1964).Generalized Functions,1, Academic Press, New York.
Herz, C.S. (1955). Bessel functions of matrix argument,Ann. Math.,61, 474–523.
Kashiwara, M. and Vergne, M. (1979). Functions on the Shilov boundary of the generalized half plane, inNoncommutative Harmonic Analysis, Springer Lecture Notes in Mathematics,728, 136–176.
Pevsner, M. (1998).Analyse conforme sur les algébres de Jordan, Thèse de doctorat, Université de Paris VI.
Springer, T.A. (1973). Jordan algebras and algebraic groups,Erg. Math.,75, Springer-Verlag, Berlin.
Strichartz, R.S. (1974). Fourier transform and non-compact rotation groups,Indiana Univ. Math. J.,24, 499–526.
Author information
Authors and Affiliations
Additional information
Communicated by Robert Strichartz
Rights and permissions
About this article
Cite this article
Clerc, JL. A generalized hecke identity. The Journal of Fourier Analysis and Applications 6, 105–111 (2000). https://doi.org/10.1007/BF02510121
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02510121