NuHAG :: TALKS

Talks given at NuHAG events

Banach Frames for Banach Gelfand Triples

  Hans G. Feichtinger (NuHAG, Faculty of Mathematics, Univ. Vienna)

  given at Palermo (17.09.19)
  id: 3673
  length: 45min
  status:
  type:

LINK-Presentation: https://nuhagphp.univie.ac.at/dateien/talks/3673_PalermoFei19B.pdf

ABSTRACT:

TeX:
The theory of {\it frames} is meanwhile very well established and
has found a number of applications, but also a huge number of
extensions and generalizations, not always with relevant examples.

The viewpoint adopted in the first lecture finds it most natural
example in the area of {\it Gabor analysis} and {\it modulation spaces}, because
in this setting there are now orthonormal bases of Gaborian type (with
good TF-concentration, due to the Balian-Low principle). Hence we have
to work with Banach frames. Formally this concept was introduced by
K.~Gr\"ochenig in \cite{gr01}, which was the natural continuation of our
joint work on {\it coorbit spaces}, a unified approach to wavelets, Gabor
expansions (and more recently shearlet and Blaschke groups, and many others).
The aim of that paper was to show the equivalence of frame conditions
(in the sense of reconstruction from a sampled continuous ``voice transform'')
and corresponding {\it atomic decomposition}, i.e. the possibility to
obtain all the elements of a given space via (non-unique) atomic representations
of its elements with control of the coefficients in the series expansion.

The main purpose of this talk will be to point out (one more time) that
it is not enough to have a Banach frame for a given Banach frame, or for
every Banach frame in a given family, which allows to identify the given
space was a closed subspace of a corresponding (typically weighted, mixed-norm)
sequence space.
%
{\bf {In order to be useful a number of additional properties have to be satisfied!}}
(which are in fact valid in the coorbit setting). We will discuss these extra
properties and why they are so important.

As a motivation we describe - on the basis of the SVD (singular value
decomposition) in linear algebra - the situation by means of commutative
diagrams, which are much more instructive than just pairs of inequalities.

More recent examples of this type have been given (among many others,
and sometimes without using the same terminology) in the literature,
let us just mention {\it generalized coorbit spaces} (as in \cite{daforastte08})
or for {\it decomposition spaces} (see \cite{vo16-2}).

% Example of citation: \cite{P1}.

\begin{thebibliography}{00}

\bibitem{daforastte08}
S.~{D}ahlke, M.~{F}ornasier, H.~{R}auhut, G.~{S}teidl, and G.~{T}eschke.
\newblock {G}eneralized coorbit theory, {B}anach frames, and the relation to
$\alpha$-modulation spaces.
\newblock {\em Proc. London Math. Soc.}, 96(2):464--506, 2008.

\bibitem{fegr89}
H.~G. {F}eichtinger and K.~{G}r{\"o}chenig.
\newblock {B}anach spaces related to integrable group representations and their
atomic decompositions, {I}.
\newblock {\em J. Funct. Anal.}, 86(2):307--340, 1989.

\bibitem{gr91}
K.~{G}r{\"o}chenig.
\newblock {D}escribing functions: atomic decompositions versus frames.
\newblock {\em Monatsh. Math.}, 112(3):1--41, 1991.

\bibitem{gr01}
K.~{G}r{\"o}chenig.
\newblock {\em {F}oundations of {T}ime-{F}requency {A}nalysis}.
\newblock {A}ppl. {N}umer. {H}armon. {A}nal. {B}irkh{\"a}user, {B}oston,
{M}{A}, 2001.

\bibitem{vo16-2}
F.~{V}oigtlaender.
\newblock {S}tructured, compactly supported {B}anach frame decompositions of
decomposition spaces.
\newblock {\em {A}r{X}iv e-prints}, dec 2016.