On the Schwartz space $$ {\mathcal {S}}(G(k)\backslash G({\mathbb {A}})) $$

On the Schwartz Space of the Basic Affine Space

Let G be the group of points of a split reductive algebraic group G over a local field k and let X = G/U where U is the group of k-points of a maximal unipotent subgroup of G. In this paper we construct certain canonical G-invariant space S(X) (called the Schwartz space of X) of functions on X , which is an extension of the space of smooth compactly supported functions on X . We show that the s...

The Schwartz space and the Fourier transform

Hubble Space Telescope Imaging of the Schwartz

Schwartz 82 (IM Lupi; HBC 605; IRAS 15528-3747) is an M0 T Tauri star located within the Lupus star forming clouds at a distance of ∼190 pc (Wichmann et al. 1998). It is one of four young stellar objects in a small CO(1-0) core near the extreme T Tauri star RU Lupi (Tachihara et al. 1996). Unlike its more famous neighbor, Sz 82 displays only a modest amount of emission-line activity, with an Hα...

Strong Divergence of Reconstruction Procedures for the Paley-Wiener Space $\mathcal{PW}^1_π$ and the Hardy Space $\mathcal{H}^1$

Previous results on certain sampling series have left open if divergence only occurs for certain subsequences or, in fact, in the limit. Here we prove that divergence occurs in the limit. We consider three canonical reconstruction methods for functions in the Paley-Wiener space PW 1 π. For each of these we prove an instance when the reconstruction diverges in the limit. This is a much stronger ...

A theorem on the Schwartz space of a reductive Lie group.

The purpose of this paper is to define the Fourier transform of an arbitrary tempered distribution on a reductive Lie group. To this end we define a topological vector space, [unk](G), in terms of the classes of irreducible unitary representations of G, which plays role of a dual Schwartz space. Our main theorem then asserts that the usual L(2) Fourier transform, when restricted to functions in...