Vol. 54, No. 2, 1974

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ISSN: 0030-8730
Closure theorems for affine transformation groups

Richard P. Gosselin

Vol. 54 (1974), No. 2, 53–57
Abstract

Let he a closed subgroup of the group of linear transformations of Rn onto itself. Let hx denote the image of the point x under the transformation h, and let 𝒢 be the transpose grotip of : i.e. its elements are associated with matrices which are the transposes of those in . For f in L2(Rn), let Cl{f;ℋ× Rn} denote the closure in the L2 norm of the linear span of functions of the form f(hx + t) where h is in , and t is in Rn. Since this space is translation-invariant, it is of the form L2(S): i.e. the set of L2 functions r(x) such that the nonzero set of r, the Fourier transform of r, is, except for a set of measure zero, included in S. In the first theorem a precise description of S is given, and in the second, a function is constmcted in a natural way whose translates alone generate the given space.

Mathematical Subject Classification 2000
Primary: 43A30
Milestones
Received: 21 February 1973
Published: 1 October 1974
Authors
Richard P. Gosselin