in Section 10]. A crucial step is the reduction of the problem to a classical result of L. Schwartz on mean periodic functions. For a modern treatment of mean periodic functions in symmetric spaces, see .Finally, in Section 5, we present some results related to the Cheeger constant (Theorem 5.1) and to the heat kernel (Theorem 5.6) of non-compact harmonic manifolds.Acknowledgements: Both authors are grateful to the University of Cyprus for the financial support. We also thank Professor Yiorgos-Sokratis Smyrlis for useful conversations. Radial eigenfunctions and convolutionsHenceforth, (X, g) denotes a non-compact, complete, simply connected harmonic space, θ(r) the density function of a geodesic sphere of radius r > 0, H ≥ 0 the mean curvature of all horospheres, and x 0 ∈ X a particular reference point. Let r(x) := d(x 0 , x). The closed ball of radius r > 0 around x ∈ X is denoted by B r (x) ⊂ X. For the inner product, we use the notationLet D(X), resp., E(X) denote the vector space of smooth functions on X, resp., smooth functions with compact support, equipped with the topology of uniform convergence of all derivatives on compact sets, see [Hel-84, Ch. II §2] for instance.

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Abstract. We show that the Cheeger isoperimetric constant of a solvable simply connected Lie group G with Lie algebra G istr (ad (H)).

We prove that spherical spectral analysis and synthesis hold in Damek-Ricci spaces and derive two-radius theorems.