# Postmodern Fourier Analysis: Reconsidering Classical Fourier Analysis from a Time-Frequency View-Point

Hans G. Feichtinger

given at  Abel Symposium 2012, Oslo (23.08.12 13:30)
id:  2292
length:  50min
status:  invited
type:
ABSTRACT:
Talk by Hans G. Feichtinger:

Postmodern Fourier Analysis:
Reconsidering Classical Fourier Analysis from a Time-Frequency View-Point

Classical Fourier Analysis has developed from Fourier series through Fourier transform
in one and several variables. In both situations Lebesgue integration plays a fundamental
role, e.g. for Fourier inversion or the verification of Plancherel's theorem. The existence
of the Haar measure and Pontrjagin's duality theory for LCA groups have laid the foundation
for Harmonic Analysis over LCA groups, as promoted in the book by A.Weil, providing
the appropriate natural framework for an abstract Fourier transform, convolution theorems
or Plancherel's theorem. Finally the theory of Schwartz tempered distributions has
allowed to extend the Fourier transform beyond the setting of functions and has made it
a crucial tool for Hoermander's approach to (pseudo-) differential operators.
A rich variety of fast codes for the DFT, the discrete Fourier transform (FFT, FFTW, etc.)
plays a crucial role in many electronic devices of our daily life, allowing the efficient
realization of algorithms for digital signal and image processing.

Modern time-frequency analysis is providing an appropriate framework for signals, functions,
or distributions which cannot be analyzed by any of the classical tools, because they assume
typically either periodicity of decay at infinity. The very natural way out of this problem
is the use of a local Fourier transforms, with a "sliding window" , the so-called STFT
(short-time Fourier transform). As a result the objects under consideration are (continuous)
functions over the so-called time-frequency plane of phase space. Among the most important
windows one finds of course the Gauss function, due to its optimal TF-concentration and
its Fourier invariance. In this way a very natural link to Fock spaces (the range of the
transform) and to coherent states is given. It is one of the crucial observations that for
any decent window the STFT can be completely recovered from its samples over sufficiently
dense (in a geometric sense) lattices. The corresponding theory is Gabor analysis, referring
to Gabor's seminal paper of 1946.

The mathematical analysis of Gabor expansions is an ongoing enterprize with many
interesting and deep mathematical results, with many interesting links between group
representation theory (mostly of the Heisenberg group), the theory of function spaces
and complex analysis methods. Robustness of Gabor expansions, the study of Gabor
multipliers, the localization theory of regular Gabor frames and many other results
require typically some mild properties on the windows under consideration, and not
just square integrability (which is OK for the continuous STFT). The (minimal TF-invariant)
Segal algebra $S_0(R^d)$ appears to be the most appropriate universal space for many
of these questions. It is also in the center of a so-called Banach Gelfand triple which
allows to describe mapping properties of classical linear mappings, e.g. the Fourier
transform, or the correspondence between linear operators and corresponding
spreading or Kohn-Nirenberg symbols and have by now an important role in the
theory of pseudo-differential operators (cf. the so-called Sjoestrand class).

As we will point out these Banach spaces (of function resp. distributions) are not only
appropriate for the description of questions arising in Gabor analysis or TF-analysis
in general, but form also a quite appropriate setting for questions of classical
Fourier analysis. Functions from $S_0(R^d)$ are appropriate summability kernels,
the dual space can be used to discuss questions of spectral analysis, and one can base
a relatively elementary theory of generalized stochastic processes on this frame-work.
Finally this setting is also well suited for a description of the problem of numerical
approximation of continuous questions (e.g. by sampling and periodization) of
continuous problems, both in Fourier analysis or TF-analysis.

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