Vol. 52, No. 2, 1974

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An analogue of the Paley-Wiener theorem for certain function spaces on SL(2, C)

Andrew Bao-hwa Wang

Vol. 52 (1974), No. 2, 617–629

The classical theorem of Paley-Wiener is concerned with characterizing Fourier transforms of C functions of compact support on the real line. It states that an entire holomorphic function F is the Fourier-Laplace transform of a C function on the real line R with support in |x|R it and only if for given integer m, there exists a constant Cm such that

|F(ξ + iη)| ≦ Cm (1 + |ξ + iη|)−m expR |η|, ξ,η ∈ R.

The purpose of this paper is to prove an analogue of this theorem for certain convolution subalgebras of C functions with compact support on the group SL(2,C), by using Fourier transform involving elementary spherical functions of general type δ.

Mathematical Subject Classification 2000
Primary: 43A75
Received: 21 June 1973
Published: 1 June 1974
Andrew Bao-hwa Wang