Always Look on the Positive-Definite Side of Life

Abstract: This thesis concerns distributions on Rn with the property of being positive-definite relative to a finite subgroup of the orthogonal group O(n). We construct examples of such distributions as the inverse Abel transform of Dirac combs on the geometries of Euclidean space Rn and the real- and complex hyperbolic plane H2, H2 C. In the case of R3 we obtain Guinand’s distribution as the inverse Abel transform of the Dirac comb on the standard lattice Z3 Ç R3. The main theorem of the paper is due to Bopp, Gelfand-Vilenkin and Krein, stating that a distribution on Rn is positive-definite relative to a finite subgroup W Ç O(n) if and only if it is the Fourier transform of a positive W-invariant Radon measure on n z 2 Cn : z 2W.z o ½ Cn . We present Bopp’s proof of this theorem using a version of the Plancherel-Godement theorem for complex commutative ¤-algebras.