ABSTRACT
Fix 0 p ^ 2 and let CP(G) be the £P-Schwartz space for G = SI(2,R). If T
denotes the operator-valued Fourier transform on G, then the image of CP(G) under T
is realized as a space of operator-valued functions defined on the union of Z - {0} with
two strips in (Z - {0} and two copies of C are the parametrizing sets for the discrete
series and the non-unitary principal series respectively). Denoting the image as CP(G),
there is a natural topology on CP(G) which makes the mapping T : CP(G) - CP(G) a
topological isomorphism. A similar theorem is obtained for C (G), the zero-Schwartz
space, defined as the intersection of all the
£P-Schwartz
spaces.
This result establishes for 5L(2,R) a result conjectured by Peter Trombi for all
semi-simple Lie groups of real rank one. The proof requires detailed analysis of the
asymptotic expansions for discrete and non-unitary principal series matrix coefficients.
Key words and phrases. Asymptotic approximation of matrix coefficients, Fourier
transform, Lp spaces, SL(2,R), Schwartz space, spherical functions.
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