An exotic minimal Banach space of functions. (English) Zbl 1019.46025

Feichtinger, Hans G.; Zimmermann, Georg

Math. Nachr. 239-240, 42-61 (2002).

Authors’ abstract “This note describes a new Banach space \(\mathcal B_0\) of square integrable functions on \(\mathbb R^d\) having many interesting invariance properties. In fact, the Fourier transform, time-frequency shifts, and \(L^2\)-normalized dilations act isometrically on it. For its definition, we make use of a general construction principle for minimal invariant spaces. We demonstrate a variety of properties following immediately from this principle. Furthermore, we give a number of different characterizations, including various atomic decompositions, as well as natural necessary and sufficient conditions for an \(L^2\)-function to belong to this new space. It turns out that this new space is somewhat exotic, since it is neither rearrangement invariant nor solid”.

Reviewer: Joan Cerdà (Barcelona)

MSC:

46E30

Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Keywords:

minimal spaces; atomic decomposition; Gabor expansions; Segal algebra