Generalized coorbit theory, Banach frames, and the relation to α-modulation spaces

AbstractCoorbit theory is a powerful machinery that constructs a family of Banach spaces, the so-called coorbit spaces, from well-behaved unitary representations of locally compact groups. A core feature of coorbit spaces is that they can be discretized in a way that reflects the geometry of the underlying locally compact group. Many established function spaces such as modulation spaces, Besov spaces, Sobolev–Shubin spaces, and shearlet spaces are examples of coorbit spaces. The goal of this survey is to give an overview of coorbit theory with the aim of presenting the main ideas in an accessible manner. Coorbit theory is generally seen as a complicated theory, filled with both technicalities and conceptual difficulties. Faced with this obstacle, we feel obliged to convince the reader of the theory’s elegance. As such, this survey is a showcase of coorbit theory and should be treated as a stepping stone to more complete sources.

Download Full-text

Banach frames for multivariate α-modulation spaces

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2005.08.091 ◽

2006 ◽

Vol 321 (2) ◽

pp. 880-895 ◽

Cited By ~ 19

Author(s):

Lasse Borup ◽

Morten Nielsen

Keyword(s):

Modulation Spaces ◽

Banach Frames

Download Full-text

Coorbit Spaces and Banach Frames on Homogeneous Spaces with Applications to the Sphere

Advances in Computational Mathematics ◽

10.1023/b:acom.0000016435.42220.fa ◽

2004 ◽

Vol 21 (1/2) ◽

pp. 147-180 ◽

Cited By ~ 34

Author(s):

Stephan Dahlke ◽

Gabriele Steidl ◽

Gerd Teschke

Keyword(s):

Homogeneous Spaces ◽

Banach Frames ◽

Coorbit Spaces

Download Full-text

Schatten p-class property of pseudodifferential operators with symbols in modulation spaces

Nagoya Mathematical Journal ◽

10.1215/00277630-1543796 ◽

2012 ◽

Vol 205 ◽

pp. 119-148

Author(s):

Masaharu Kobayashi ◽

Akihiko Miyachi

Keyword(s):

Pseudodifferential Operators ◽

Modulation Spaces

AbstractIt is proved that the pseudodifferential operators σt(X, D) belong to the Schatten p-class Cp, 0 < p ≤ 2, if the symbol σ(x,ω) is in certain modulation spaces on

Download Full-text