# The role of Banach Gelfand Triples in the context of Conceptual Harmonic Analysis

Hans G. Feichtinger

given at  Linne University, Vaexjoe, ISAAC2017 (16.08.17 09:30)
id:  3323
length:  50min
status:  invited
type:
ABSTRACT:
The idea of Conceptual Harmonic Analysis'' grew out of the attempt to make objects arising in Fourier Analysis or Gabor Analysis (such as norms of functions, their Fourier transforms, dual Gabor atoms, etc.) computable. Using suitable function spaces such as the Segal algebra $S_0(R^d)$ it should be possible to find concrete algorithms which allow to compute approximations on existing hardware in finite time, up to (at least potentially) arbitrary requested precision.

Going beyond the ideas of Abstract Harmonic Analysis, which only allows to identify the analogies between objects on different LCA (locally compact Abelian) groups, the idea of Conceptual Harmonic Analysis is to emphasize the connections between these settings, for example, in order to use methods from discrete, periodic Gabor analysis (computationally realizable using e.g.\ MATLAB) in order the study
the continuous case.

The Banach Gelfand Triple $(S_0,L^2,S_0')({R}^d)$, which can also be seen as a {\it rigged Hilbert space''}, justifies a number of such procedures providing a tool to deal with periodic, or continuous or discrete signals in a unified way. As it turns out the so-called $w*$-convergence is crucial, and fine partitions of unity as well as regularizing operators play a crucial role in this context. Due to the {\it kernel theorem} for this setting also various spaces of operators can be studied.

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% BIBLIOGRAPHY (can be omitted)
\noindent
[1] Elena~{C}ordero, Hans~G. {F}eichtinger, and Franz~{L}uef.
\newblock {B}anach {G}elfand triples for {G}abor analysis.
\newblock In {\em {P}seudo-differential {O}perators}, Vol.\ 1949 of {\em
{L}ecture {N}otes in {M}athematics}, {S}pringer (2008), 1--33.
\\[1ex]
[2] Hans G. Feichtinger, \emph{{E}lements of {P}ostmodern {H}armonic {A}nalysis.}
\newblock In {\em {O}perator-related {F}unction {T}heory and {T}ime-{F}requency
{A}nalysis. {T}he {A}bel {S}ymposium 2012, {O}slo, % {N}orway,
{A}ugust, 2012}, {S}pringer (2015), 77--105.

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