This paper is intended as an introduction to the Abel transform for the non·· specialist. Nevertheless it contains some of the essential ideas which enables us to present some recent results on this transform in the last section.

The Abel transform plays an important role in the theory of the spherical Fourier transform on synm1e'tric spaces of 'the noncompac't 'type. Its role is analogous t:o the role of 'the Radon 'transform in the theory of the ordinary Fourier transform on IRn. Therefore we first present the example of the ordin.ary Fourier 'transform in sec'tion l. Then we 'turn to the of SL(2,1R) (sections 2 and 3). Here we give an explicit expression for H'le Abel transform and review some of the results and applications. This will serve as motivation and as prototype for further research. In the last section we present some recent results.

In the unimodular case, the Frobenius reciprocity theorem for irreducible square integrable representations asserts that certain intertwining spaces are canonically isomorphic; the essential analytic point is that square integrability implies the continuity of functions in particular subspaces of $L^2$ spaces on which the group acts and leads to a characterization of these subspaces in terms of reproducing kernels. In the nonunimodular case this is no longer true. There is a canonical isomorphism between proper subspaces of the intertwining spaces, one of which is uniformly dense in the full intertwining space.

In their paper on the Szegö map, Knapp and WaJlach [KW] remarked that in the case of those discrete series representations of SU(2, 1) which occur as the subquotient of three principal series representations, their methods provided only two of these. That is, the Szegö map built from the highest weight vector misses some occurrences of discrete series represen- · tations as quotients. In this note we use the extension of Szegö maps due to Gilbert, Stanton, Kunze, and Tomas [GKST] to investigate these further cases.

An explicit description is given of a real-valued function $f$ on 2 $[- \pi, \pi]$ which is zero in a neighbourhood of 0 but for which the square partial Fourier sums $S_n f$ satisfy $lim$ $sup_n S_n f(0,0) = \infty$. Furthermore, the function is infinitely differentiable everywhere except along the y-axis where it is continuous. Also its support is contained in a square at dis·tance $\pi/2$ from 0 and the square may be chosen to have arbitrarily small sides. Finally, neither of the axes intersect the interior of the support of $f$.

The first part of this article is an introduction to uncertainty principles in Fourier analysis, while the second summarizes some recent work by the authors and also by Michael Cowling and the authors.

The following (rather vague) principle is well known to every student of classical Fourier analysis: If a function $f$ is 'concentrated' then its Fourier transform $f$ is 'spread out' and vice-versa. After reviewing three precise (and different) formulations of this principle in classical Fourier analysis on $R^n$, we will describe how it extends to LCA groups and certain nonabelian Lie groups - for instance, semisimple Lie groups and Heisenberg groups.

Let G denote a compact abelian group and B a Banach algebra of continuous functions defined on G with pointwise multiplication. G.E. Silov called B of type C if its norm is equivalent to that defined by \[ \|b\|^c = sup \hspace{0.1in} inf\{\|c\|_B : c \in B\hspace{0.1in} ,\hspace{0.1in} c(x) = b(x)\}\hspace{0.1in} ,\hspace{0.1in} x \in G \] and gave a complete classification of those algebras which are homogeneous and of type C. In this paper, we first replace pointwise multiplication by convolution, before generalizing the notion of type C to homogeneous Banach spaces. Again a complete classification is obtained.