On Multiplicative Maps of Continuous and Smooth Functions

Details On Multiplicative Maps of Continuous and Smooth Functions On Multiplicative Maps of Continuous and Smooth Functions

2011-11-20

arxiv.org/abs/1111.4658v1

Mathematics

Classical Analysis and ODEs

Accepted for publication in GAFA Seminar Notes

Scientific paper

In this note, we study the general form of a multiplicative bijection on several families of functions defined on manifolds, both real or complex valued. In the real case, we prove that it is essentially defined by a composition with a diffeomorphism of the underlying manifold (with a bit more freedom in families of continuous functions). Our results in the real case are mostly simple extensions of known theorems. We then show that in the complex case, the only additional freedom allowed is complex conjugation. Finally, we apply those results to characterize the Fourier transform between certain function spaces.

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