Explicit representations of spaces of smooth functions and distributions

Abstract

We provide an explicit isomorphism between the space of smooth functions E(Rd) and its sequence space representation s⊗ˆCN which isomorphically maps various spaces of smooth functions onto their sequence-space representation, including the space D(Rd), of test functions, the space of Schwartz functions S(Rd) and the space of “p-integrable smooth functions” DLp(Rd). By restriction and transposition, this isomorphism yields an isomorphism between the space of distributions D′(Rd) and its sequence space representation s′⊗ˆπCN which analogously maps various spaces of distributions isomorphically onto their sequence space representation. We use this isomorphism to construct both a common Schauder basis for these spaces of smooth functions and a common Schauder basis for the corresponding spaces of distributions.