1,086 research outputs found
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
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