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For G a compact separable Hausdorff topological group and for 1 0 (a E G), and (iv) f(xy) = f(yx) (x, y E G), then f E Lq(G, dv') where q = p/(p 1). Even more striking in view of other possible proofs and the history, is the elegance and simplicity of their argument. Subsequently, Kawazoe and Miyazaki, using the method in [1], proved Theorem A for compact semisimple Lie groups G and an analogous result for spherical Fourier decompositions of K-bi-invariant functions on compact symmetric spaces (see Theorem B) [6]. It is the aim of this note to observe that by using the Peter-Weyl theory, the proofs in [6] apply for arbitrary separable compact Hausdorff topological groups in the case Received by the editors March 25, 1991. 1991 Mathematics Subject Classification. Primary 43A30, 43A1 5. ? 1993 American Mathematical Society 0002-9939/93 $1.00 + $.25 per page

Fourier decompositions with positive coefficients in compact Gel′fand pairs