On the Fourier transform of Schwartz functions on Riemannian Symmetric Spaces

Consider the (Helgason-) Fourier transform on a Riemannian symmetric space G/K. We give a simple proof of the L^p-Schwartz space isomorphism theorem (0 <p \le 2) for K-finite functions. The proof is a generalization of J.-Ph. Anker's proof for K-invariant functions

Elementary proofs of Paley-Wiener theorems for the Dunkl transform on the real line

We give an elementary proof of the Paley-Wiener theorem for smooth functions for the Dunkl transforms on the real line, establish a similar theorem for L^2-functions and prove identities in the spirit of Bang for L^p-functions. The proofs seem to be new also in the special case of the Fourier transform.Comment: 9 pp., LaTeX, no figures; final version, to appear in Int. Math. Res. No

Real Paley-Wiener theorems and local spectral radius formulas

We systematically develop real Paley-Wiener theory for the Fourier transform on R^d for Schwartz functions, L^p-functions and distributions, in an elementary treatment based on the inversion theorem. As an application, we show how versions of classical Paley-Wiener theorems can be derived from the real ones via an approach which does not involve domain shifting and which may be put to good use for other transforms of Fourier type as well. An explanation is also given why the easily applied classical Paley-Wiener theorems are unlikely to be able to yield information about the support of a function or distribution which is more precise than giving its convex hull, whereas real Paley-Wiener theorems can be used to reconstruct the support precisely, albeit at the cost of combinatorial complexity. We indicate a possible application of real Paley-Wiener theory to partial differential equations in this vein and furthermore we give evidence that a number of real Paley-Wiener results can be expected to have an interpretation as local spectral radius formulas. A comprehensive overview of the literature on real Paley-Wiener theory is included.Comment: 27 pages, no figures. Reference updated. Final version, to appear in Trans. Amer. Math. So

Local spectral radius formulas for a class of unbounded operators on Banach spaces

We exhibit a general class of unbounded operators in Banach spaces which can be shown to have the single-valued extension property, and for which the local spectrum at suitable points can be determined. We show that a local spectral radius formula holds, analogous to that for a globally defined bounded operator on a Banach space with the single-valued extension property. An operator of the class under consideration can occur in practice as (an extension of) a differential operator which, roughly speaking, can be diagonalised on its domain of smooth test functions via a discrete transform, such that the diagonalising transform establishes an isomorphism of topological vector spaces between the domain of the differential operator, in its own topology, and a sequence space. We give concrete examples of (extensions of) such operators (constant coefficient differential operators on the d-torus, Jacobi operators, the Hermite operator, Laguerre operators) and indicate further perspectives.Comment: Minor changes in presentation. 23 pages, final version, to appear in Journal of Operator Theor