Björn Walther:On the Finite Dimensionality of Spaces of Absolutely Convergent Fourier Transforms

Abstract. Since the space of absolutely convergent Fourier transforms is dense
in the space of continuous functions which vanish at infinity (I.E.
Segal 1950) sufficiently many functions are Fourier transforms. On the
other hand, the space of absolutely convergent Fourier transforms
either coincides with the space of continuous functions which vanish
at infinity or is of the first category in that space (S. Banach
1932). Hence very few functions are absolutely convergent Fourier
tranforms.

A further indication of the absolutely convergent Fourier transforms
being very few is the following result of K. Karlander (Math. Scand.
1997): Let Y be a closed subspace of the space of continuous functions
on the real line which vanish at infinity. Assume that all elements in
Y are absolutely convergent Fourier transforms. If in addition Y is
reflexive, then Y is of finite dimension.

The purpose of this presentation is to discuss on this result and its
extensions.