For locally compact, second countable, type I groups G, we characterize all closed (two-sided) translation invariant subspaces of L²(G). We establish a similar result for K-biinvariant L²-functions (K a fixed maximal compact subgroup) in the context of semisimple Lie groups.

Let G be a compactly generated, locally compact group with polynomial growth and let ω be a weight on G. We look for general conditions on the weight which allow us to develop a functional calculus on a total part of L(G,ω). This functional calculus is then used to study harmonic analysis properties of L(G,ω), such as the Wiener property and Domar's theorem.

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