Let G be a compact nonfinite metric group and let \(\hat G\) be its unitary dual. Suppose \(f\in L^ 1(G)\) and E is a finite subset of the dual. Let \(\hat f\) be the Fourier transform of f. Suppose that w(x) is the area of the ball in G with center e and x on its boundary. Then conditions are given for \(\alpha\),\(\beta\in {\mathbb{R}}\) so that there is a positive constant \(K=K(\alpha,\beta)\) such that: \[ \sum_{\tau \in E}d(\tau)Tr(\hat f(\tau)\hat f(\tau)^*)\leq K(\sum_{\tau \in E}d(\tau)^ 2)^{\alpha}\| w^{\beta}f\|_ 2 \] for all such E and f. Here d(\(\tau)\) is the degree of the representation \(\tau\). The paper ends with a less complete analogue for compact Riemannian manifolds and considers the 2-dimensional sphere in detail.
This paper generalizes a previous paper by J. F. Price and P. C. Racki [Proc. Am. Math. Soc. 93, 245-251 (1985; Zbl 0562.42015)] which considered the case of multiple Fourier series. Related results for Euclidean space were obtained by W. G. Faris [J. Math. Phys. 19, 461-466 (1978)] and J. F. Price [Stud. Math. 85, 37-45 (1986; Zbl 0636.46029)]. These inequalities are rather different from the classical uncertainty inequality for \({\mathbb{R}}\) as described in H. Dym and H. P. McKean, Fourier Series and Integrals (1972; Zbl 0242.42001). This result does not directly generalize to \({\mathbb{R}}/{\mathbb{Z}}\) since the obvious generalization fails for constants. F. A. Grünbaum has noticed, however, that one can find an analogue for periodic functions of period 1 such that \(f(0)=f(1)\). Another sort of uncertainty principle for the symmetric space of the Lie group \(G=SU(p,q)\) is described by M. Shahshahani [Am. J. Math. 111, 197-224 (1989)].