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Banach Gelfand Triples and Rigged Hilbert Spaces


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

  given at  Palermo (16.09.19)
  id:  3672
  length:  45min
  status: 
  type: 
  LINK-Presentation:  https://nuhagphp.univie.ac.at/dateien/talks/3672_PalermoFei19A.pdf
  ABSTRACT:

  TeX:
The purpose of this talk is to point out, that very often
{\it Hilbert spaces $\Hilb$ are surrounded by a family of similar function
spaces} of a similar nature. It also occurs quite often, that
the intersection of all these spaces forms itself a topological
vector space, let us write $\Scsp$ , often called a space of {\it test functions}, whose
dual (the space of continuous linear functionals on this space
of test functions, its elements are often called distributions
or ultra-distributions) contains not only the Hilbert space but
also all the ``surrounding'' Banach spaces. We write $\ScPsp$.
The triple $(\ScGTr)$ is then called a Gelfand Triple.

For the case of the Euclidean space on can start with $H = L^2(R^d)$
and consider the collection of all (polynomially) weighted $L^p$-spaces
as well as the corresponding Sobolev space $H^s(R^d)$. Their intersection
gives the classical Schwartz space $\Scsp(R^d)$ of rapidly decreasing
functions on $R^d$, with the dual space $\ScPsp(R^d)$ of {\it tempered
distributions}. The whole class of Besov-Triebel-Lizorkin spaces
is contained between $\Scsp(R^d)$ and $\ScPsp(R^d)$. This is what makes
up the ``theory of function spaces'' in the spirit of H.~Triebel.

% \end{document}

Here as well in many other situations one can describe the space
of test functions naturally as the common domain of the
powers/iterates of some unbounded (self-adjoint) operator on $\Hilb$.
In the concrete case one might take the Schr\"odinger operator and
describe all some of the relevant spaces, including the so-called
Shubin classes, via Hermite expansions and the corresponding
decay resp.\ growth of the corresponding coefficients. This is
usually the way to prove a so-called {\it kernel theorem}, which
allows to characterize the vector space of all continuous linear
operators from $\Scsp$ into $\ScPsp$ by (distributional integral)
kernels in $\ScPsp$ (over the product domain).

As we know this concrete Banach Gelfand triple also has a nice
description in terms of wavelet expansions and in particular
using Gabor expansions. In fact, replacing the
continuous wavelet transform (CWT) by the STFT (short-time
Fourier transform) one ends up by looking at the more
and more used family of {\it modulation spaces} (introduced in
1983), among them the now classical spaces $M^s_{p,q}(R^d)$, among the
unweighted modulation spaces $M_{p,q}(R^d)$.

By focussing on the simple {\bf Banach Gelfand Triple}
% $\SOGTrRd$
$(S_0,L_2,S_0')(R^d)$,
(with $S_0(R^d) = M^1(R^d) = M_{1,1}(R^d)$ and $S_0'(R^d) = M_{\infty,\infty}(R^d)$)
we have a technically much more simple situation, which allows
to explain a number of concepts using only basic concepts from
functional analysis, including Banach spaces, bounded linear
functionals and operators, and the $w^* $-topology.

% \end{document}

We will focus on a couple of important results that can be
obtained in this way, in particular the role of the perhaps
somewhat surprising existence of a kernel theorem (despite
the obvious absence of any nuclear Frechet space).

Within classical Fourier analysis the first such Banach
Gelfand Triple was introduced in principle by N.~Wiener,
when he used $A(T)$, the algebra of absolutely convergent
Fourier series, sitting inside $\Hilb = L^2(T)$, and contained
in the space $PM$ of pseudo-measures. The identification % PMTN does not exist
of this BGTr with the {\it canonical} BGTr $(\ell^1,\ell^2,\ell^\infty)$
can be realized
by extending the classical Fourier series expansion (e.g. using
duality, or distributional Fourier coefficients). In some sense
the discussed BGTr allows to transfer this situation to the
setting of general LCA groups.


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