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We systematically develop real Paley-Wiener theory for the Fourier transformon R^d for Schwartz functions, L^p-functions and distributions, in anelementary treatment based on the inversion theorem. As an application, we showhow versions of classical Paley-Wiener theorems can be derived from the realones via an approach which does not involve domain shifting and which may beput to good use for other transforms of Fourier type as well. An explanation isalso given why the easily applied classical Paley-Wiener theorems are unlikelyto be able to yield information about the support of a function or distributionwhich is more precise than giving its convex hull, whereas real Paley-Wienertheorems can be used to reconstruct the support precisely, albeit at the costof combinatorial complexity. We indicate a possible application of realPaley-Wiener theory to partial differential equations in this vein andfurthermore we give evidence that a number of real Paley-Wiener results can beexpected to have an interpretation as local spectral radius formulas. Acomprehensive overview of the literature on real Paley-Wiener theory isincluded.

Real Paley-Wiener theorems and local spectral radius formulas