An Exotic Minimal Banach Space of Functions

✍ Scribed by Hans G. Feichtinger; Georg Zimmermann

• Publisher: John Wiley and Sons
• Year: 2002
• Tongue: English
• Weight: 299 KB
• Volume: 239-240
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/1522-2616(200206)239:1<42::aid-mana42>3.0.co;2-#

✦ Synopsis

This note describes a new Banach space B 0 of square integrable functions on 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.

2000 Mathematics Subject Classification. 46E30.