Translation-invariant linear forms on L2(G) for compact Abelian groups G
The first author has proved previously [J. Functional Analysis8 (1971), 173–188] that translation-invariant linear forms on (Rn), as well as on several other spaces of C∞ test functions and distributions, are necessarily continuous. The analogous result is proved here (Theorem 3) for the Hilbert space L2(G), where G is a compact Abelian group with a finite number of components, as well as for a one-parameter family of spaces p of pseudo-measures on G, for 1 ⩽ p < ∞. These results are new even for the case that G is the circle group T. The facts (Theorem 1) underlying these new results turn out to be somewhat different from the corresponding facts underlying the case of (Rn). This is reflected in the method of proof which is rather different here from that [ibidem] for the previous cases, even though both methods make full use of the characterization of the Fourier transforms of the objects (functions, pseudo-measures, or distributions) under consideration.