Harmonic Analysis

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Elements of Postmodern Harmonic Analysis
Hans G. Feichtinger
[email protected]
October 1, 2013
Abstract
Harmonic Analysis (HA) has the Fourier transform, convolution operators,
translation invariant linear systems, and the Shannon sampling theorem based on
Poisson’s formula as central topics. During the last century HA has split into
an abstract direction, dealing with functions over locally compact Abelian (LCA)
groups and on the other hand a variety of application areas, such as digital signal
and image processing or mobile communication, influencing our daily life.
This article describes a line of thoughts which has been developed by an ab-
stract harmonic analyst who has converted to an application oriented harmonic
analyst in the last decades. The author also likes to run numerical experiments in
order to gain insight into both practical question as well as abstract concepts.
The change of perspectives that took place by this process is described as a
transition from Classical Fourier Analysis to Postmodern Harmonic Analysis, with
strong functional analytic aspects, but also incorporating concepts from abstract
and computational harmonic analysis.
This is just a first attempt, with a modest philosophical background, but we
hope that the terminology, and above all the views, recommendations and methods
described in our this note will help to enrich harmonic analysis and contribute to
its healthy development in the years to come.
Readers who don’t agree, or support the author’s view-points expressed in this
note are encouraged to contact him and to share their views, also with younger
and older colleagues active in the field, as well as engineers or computer scientists
and other “practical users” of Fourier analysis.
Since we will talk a lot about concept and ideas we have to acknowledge that
concepts emerge in the scientific community, typically have a life span
1
where
they receive high attention, and once they have been explored properly find their
way into the canon of teaching at the universities or get discarded. Among the
concepts relevant for analysis the ideas of Fourier certainly play a central role and
have influenced the field for centuries now. Nevertheless, the chapter should not
be closed, the way how Fourier analysis is taught may need some reconsideration,
and this article will propose some ideas in such a direction.
1
See the interesting article by Wilder [62].
1
1 Mathematical Problems and their Tools
1.1 A coarse Tour d’Horizon
Let us first quickly try to summarize the development of Fourier analysis. Where did the
theory start, and how far has Fourier analysis come in the two centuries, in particular
during the last decades, where the distributional view-point appearing in the context of
micro-local analysis works with sophisticated arguments from so-called “hard analysis”.
On the other hand the numerical practice performed by people working in electrical
engineering, signal and image processing or computer science
2
is perceived nowadays as
a far distant subject having little in common with Classical Fourier Analysis, apart from
the name. In this situation we will try to argue that there is more to be said and more
to be done for the (re-)unification of field, with a high potential for synergy, making the
two branches more supportive to each other in a significant way.
Not surprisingly our story starts about 200 years ago with J.B. Fourier’s original
claim that “every periodic function” has a “representation as an infinite trigonometric
series”. It took a century to give a proper meaning to this statement, because it required
subsequent generations of mathematicians (including Riemann and Lebesgue) to develop
proper notions of what a function is (leading to the currently widely accepted concept
of a mapping) and how an integral of such a function should be interpreted, in the most
general setting. Nowadays these concepts appear as properly determined and granted,
and a person raising doubts about such well established concepts has a difficult position.
But if we try take a naive (not to say child-like) viewpoint, we have to ask ourselves: are
equivalence classes of measurable functions modulo null-functions in the Lebesgue sense
really such natural objects? Are they suitable to describe something like the change of
temperature in a room or in a town?
Even more provocatively, let us shortly discuss a very simple (physical) function on
R
4
, namely the temperature in a room or in some region, say town. One may expect that
this is a smooth function defined everywhere at first sight, describing the temperature
at each point in space using a world coordinate system and for any given moment in
time (during the observation interval). But does it really have a physical meaning to
even think of the temperature during a millisecond in a point-like volume? Is there a
clear meaning to the expression: “temperature at a given location at a particular time”,
or do we realistically only infer on the existence of such a function only indirectly (is it
a genearlized function) by measuring local and temporal means? Does it help us in this
situation that we are allowed to ignore the values of the “function” over a null-set? Will
it be an L
p
-function? In addition we have to face that computers will never be able to
handle an uncountable amount of information.
Of course this example is mentioned in order to stimulate a rethinking of the ques-
tion, whether mathematicians have discovered a god-given truth and the right way of
describing functions (under any circumstances), or whether this is more a history based,
sophisticated and universally accepted way of describing things, which might be also
replaced by alternative ways? So what about the idea of taking functions (as we think
of them now) as possible limits (available only under favorable conditions) of a natural
2
Of course we hope to convince at least some readers that the distributional view-point is not so
complex and should be further developed to be more useful for the applied scientists!
2
concept of “local average information” (nowadays called distributions). In other words,
could one think of distributions as natural objects, and difficulties in Fourier analysis
partially stemming if one is pushing inappropriately for domains (such as L
1
(R
d
))?
Of course a lot of positive things can (and should) be said about the Lebesgue
integral. In some sense the original problems (arising from questions of the convergence
of Fourier series) of Fourier theory could be elegantly answered by the theory of Lebesgue
integration, and thus younger generations learning the subject (including the author
when he was young) view this this (important) approach as being carved in stone, as
the right way to view things, as the final insight. As a consequence the way how most
of us are teaching the subject is not too different from how it was done 60 years ago,
just a little bit more elegant and with a slightly stronger functional analytic flavor.
Coming back to the Fourier transform. Once the Lebesgue integral was established
it was possible to properly define the Fourier transform on the Banach space L
1
(R
d
),
in fact on

L
1
(G), | |
1

, for any LCA group G, as soon as the existence of the Haar
measure was shown. For L
1
(G) also the convolution of two functions makes complete
sense, using Fubini’s theorem. Combining these to fundamental facts one can derive the
all important convolution theorem for the Fourier transform. In the form of characteristic
functions of a probability measure this set of tools also allows to prove e.g. the central
limit theorem, claiming that the probability distributions of averaged partial sums of
identically distributed random variables are convergent, in a suitable (weak!) sense to
the Gauss function, the density of the normal distributions.
Starting from the L
1
−theory one can even extend the Fourier transform, using the
usual completion argument, to a unitary mapping from L
2
(G) to L
2
(

G). But it is
although clear that this Fourier-Plancherel transform is not anymore an “integral trans-
form” in the strict (pointwise a.e.) sense. Furthermore Plancherel’s theorem can be
proved in a similar way starting from any reasonable dense subspace, one does not need
L
1
(R
d
) for this purpose.
If we look back we can see that Fourier Analysis was deeply involved in the de-
velopment of modern functional analysis and measure theory, the proof of Plancherel’s
theorem being a typical example of “how things can be done” if technical problems arise.
The method consists in reducing the problem to the analysis using dense subspaces and
then extend the operators, using estimates for their mapping properties with respect to
appropriate norms, using an approximation argument.
But of course the development did not stop, and new view-points have been developed
to cope with “objects” (then called generalized functions or distributions) which should
have Fourier transforms as well, somehow as limits (but in which sense?) of ordinary
functions, very much like real numbers are considered as limits of their finite decimal
approximations. And of course such considerations, if they have to be put on solid
ground, have to deal with the problem of “existence” and “natural embedding” of the
old objects (here ordinary functions) into the larger context.
The reader will easily guess that with my arguments given above I am opening
the stage for distributions, which can be elegantly described using their actions as linear
functionals on the test functions (I would call this the modern view-point). Alternatively
(but less elegant) they can be described as by a more elementary view-point working
with sequences of ordinary functions. Such an approach was prominently pursued in the
work of M.J. Ligthhill: [43]. Although it is nice to know that such an approach exists,
3
the influence of this approach on the community has been minor in the last decades
and virtually nobody is following this approach anymore, because it is too cumbersome
compared with the more powerful and elegant functional analytic setting (see e.g. [34]).
Among the different possible setting (including ultra-distributions and Gelfand-Shilov
classes) that one can choose in order to describe generalized functions the approach pro-
vided by Laurent Schwartz is certainly the most popular and in some sense the most
natural one. It is based on the space S(R
d
) (of rapidly decreasing functions) as a space
of test functions, endowed with a natural family of seminorms. The vector space of
so-called tempered distributions is then simply the dual S

(R
d
), the space of continuous
linear functionals on S(R
d
). The construction is carried out in such a way that S(R
d
)
is invariant under the Fourier transform, which allows to extend the action of the FT
to the dual space. This approach has finally revolutionizing the treatment of PDEs (see
the fundamental work of Lars H¨ ormander, [34–36], which we cannot elaborate here),
leading to micro-local analysis and the theory of pseudo-differential operators.
For questions of abstract HA over LCA groups a corresponding theory was proposed
by Bruhat (see [5, 50, 51]), but as it is using structure theory it is a highly complicated
tool and has found relatively little use in the past (see e.g. in the work of M. Rieffel [53],
or A. Vourdas [59]).
In the last two or three decades of the last century time-frequency analysis and in
particular Gabor analysis received a lot more of attention (in parallel with the rapid
growth of wavelet theory) and new methods had to be developed in order to properly
describe the situation encountered in this setting. In fact, it was not really possible to
save Gabor’s original approach by an appropriate use of distribution theory ( [38]). In-
stead, it became more important to have a family of suitable function spaces to describe
mapping properties of relevant operators, such as frame operators or Gabor multipli-
ers, namely modulation spaces (see [14] or [16]). We also had to learn how to handle
redundant representations, now known as Gabor frames (as a counterpart to Gaborian
Riesz bases). This is in contrast to the situation of wavelet theory where it was possi-
ble to design suitable orthonormal bases of wavelet type, which at the same time also
form unconditional bases for a whole range of function spaces, including the Besov- and
Triebel-Lizorkin spaces.
Interestingly enough (but not yet fully accepted or known in the community) the
tools (test functions and distributions) arising in this context are also useful in the
description of questions outside time-frequency analysis, in particular in the context
of classical Fourier analysis (summability methods) as well as for the description of
questions of discretization. In fact, one of the main points of this little note is to try to
indicate the possible use of what we call the Banach Gelfand Triple based on the Segal
algebra S
0
(G) for a wide range of problems, including Gabor analysis, but not at all
limited to this setting (see [9] for a full account).
Remark 1. We also advocate the view-point that the development of mathematics as
a discipline is not only based on the amount of facts accumulated, or on the increased
complexity of statements possible within the theory or continued specialization. On the
contrary: Major progress is based on the lucky moments in the history of our science,
where new view-points allow to get a better and easier understanding of what has been
done so far, as well as (ideally) enabling us to answer questions that could not be
answered using the “classical tools” (see corresponding remarks in [7]).
4
1.2 Classical Fourier Analysis
A good overview of what is nowadays perceived is Classical Fourier analysis is provided
by the survey talk held by Charles Fefferman at the International Mathematical Congress
1974, entitled ”Recent progress in Classical Fourier Analysis” [12]. It features among
others Calderon-Zygmund operators, H
p
-spaces, atomic decompositions and Cotlar’s
Lemma, and of course Carleson’s famous Acta paper [6], on the convergence and growth
of partial sums of Fourier series. In fact, in this period the school around E. Stein
developed systematically ways to describe the smoothness of functions, using Bessel
potentials, allowing to describe fractional smoothness, with Sobolev spaces arising as
the natural cornerstones for integer smoothness, positive or negative, as well as the
family of Besov spaces B
s
p,q
(R
d
). They can be be viewed as generalized Lipschitz spaces
with respect to L
p
-norms (see Stein’s book on Singular Integrals and Differentiability
Properties of functions [55]). See the books by Muscalu and Schlag [49] for a recent
account in this direction, or the books by Grafakos (e.g. [31]).
It turned out that the Paley-Littlewood decompositions of these spaces allow to
provide atomic decompositions of tempered distributions with the extra property, that
the (weighted) summability conditions of the corresponding coefficients (which are not
uniquely determined) in terms of weighted mixed norm spaces allow to determine the
membership of a given function in one of these smoothness space. As a typical reference
see the early work of Frazier and Jawerth ( [29]), which was inspired by the work of
Peetre. The atomic decompositions described there can in fact be seen as precursors to
modern wavelet theory.
For the pioneers in interpolation theory, Jaak Peetre and Hans Triebel, these “Func-
tion Spaces” and the use of dyadic decompositions on the Fourier transform side were
the starting point to identify the corresponding interpolation spaces, using either real or
complex interpolation methods (for pairs of Banach spaces). In this way the family of
Triebel-Lizorkin spaces F
s
p,q
(R
d
) arose (with Bessel potential spaces being special cases,
for q = 2 within this family). For a systematic summary of all the known properties
of these spaces (duality, embedding, traces and much more) the reader may consult the
books of H. Triebel (e.g. [56, 57]). It was also recognized that the L
p
-spaces belong to
this family, but only for 1 < p < ∞, while one should replace L
1
(R
d
) and L

(R
d
) by the
Hardy space H
1
(R
d
) and its dual, the famous BMO-space, respectively. Given the use-
fulness of these function spaces for many purposes relevant at that time, e.g. the study
of maximal functions or for the theory of Calderon-Zygmund (CZ) operators was giving
this line of research (among others) great visibility within the analysis community.
Later on the good fit between wavelet orthonormal systems and this setting in combi-
nation with the existence of efficient numerical algorithms was among the various reasons
why wavelet theory “took off very quickly” and in fact was immediately catching the
attention of pure and applied mathematicians and applied scientists such as electrical
engineers of computer scientists interested in image processing. From the very begin-
ning, e.g. in the first preprints of Yves Meyer, it has been pointed out that the wavelet
ONBs are not just orthonormal bases for L
2
(R
d
), but they are also unconditional bases
for all those function spaces mentioned above. Moreover, the matrix representation of
CZ-operators in such wavelet bases had good off-diagonal decay, which in turn explains
many of their good properties, and why these function spaces are very suitable for the
5
description of CZ-operators. It also became clear that good wavelets have to have good
decay (concentration in the time-domain) as well as satisfy a few moment conditions,
in order to allow the characterization of functions resp. tempered distributions through
the size of their wavelet coefficients or equivalently the membership of the continuous
wavelet transform in a suitable weighted mixed-norm space over scale space.
The intensive study of the connection between membership of functions in various
function spaces (of high or low smoothness, with or without decay in time) and the corre-
sponding wavelet coefficients, which more recently is carried over also to the anisotropic
setting, led to the insight that strong decay allows the guarantee that most of the func-
tion (e.g. in the sense of some L
p
-norm) is already encapsulated in a relatively small
number of wavelet terms. From there the theory of sparsity took off, with the search for
efficient algorithms to find the best approximation of a signal by finite partial sums of
wavelet series. Such ideas are behind certain data compression methods, using thresh-
olding methods, but also provide a setting for compressed sensing.
Although wavelet theory isn’t by far providing universal tools for analysis (nor does
any other theory), it was nevertheless very inspiring for a number of developments in
modern analysis. Describing function spaces by suitable (orthogonal and even over-
complete and non-orthogonal) sets of specific functions became an important branch of
analysis, contributing to image compression applications, to the description of pseudo-
differential operators, or the characterization of function spaces. By now we have a
theory of shearlet expansions, as well as a variety of concepts for atomic decompositions
and again the characterization of various function spaces in the complex domain.
Out of this rich variety of functional spaces, which in some respect are all very similar
to each other (the corresponding unified coorbit theory has been developed in the late
80’s, see [17]), we will pick out a few spaces which arose originally in the context of time-
frequency analysis. There is also a natural link to the Schr¨ odinger representation of the
reduced Heisenberg groups (from a group representation theoretical point of view), or
on the other hand simple to a characterization of function spaces through the behavior
of the STFT (short-time Fourier transform) of its elements (see [18]).
There are many reasons for going with this specific setting: Above all it is simple
to explain, and treats the time- and the frequency variable at equal footing, hence
it is optimally suited for a description of the Fourier transform (as a transition from
pure frequencies to Dirac point measures). Secondly it has many applications in the
description of real analysis problems (e.g. Poisson’s formula or the Fourier inversion
theorem). Finally one can obtain in this setting a kernel theorem and other useful
description of operators which reduce to MATLAB implementable terms in the case of
the group G = Z
n
.
The rest of the manuscript is organized as follows: First we give a quick summary
of the ingredients describing the Banach Gelfand Triple (S
0
, L
2
, S
0

), just enough to
indicate that is is the proper setting for the description of time-frequency analysis,
many aspects of classical analysis, even for problems in abstract harmonic analysis, but
above all a suitable format to related the connections between the different settings
(in the spirit of Conceptual Harmonic Analysis). We then provide a few typical cases
where this setting makes the description much easier and transparent than the classical
approaches found in the literature. Finally we will indicate (shortly only) the relevance
and natural occurrence of the so-called w

-convergence in the dual space S
0

(G). We list
6
a few occurrences, and how it can be used to turn typical heuristic arguments used in
the literature to solid mathematical statements formulated in a distributional setting.
2 The Idea of Conceptual Harmonic Analysis
Let us try to describe our aim and vision of “Postmodern Harmonic Analysis” through
the concept of Conceptual Harmonic Analysis, which has been proposed by the author
already a while ago, as a counterpart to abstract and applied resp. computational har-
monic analysis. Let us therefore briefly summarize the short-coming of the current AHA
view-point. Again, the historical perspective may help to understand the situation a bit
better.
2.1 From Fourier to the FFT and back
Many books in the field of Fourier Analysis follow the historical path, reminding the
reader of the establishment of the theory of Fourier series, which in the context of func-
tional analysis is nothing else than the expansion of elements in a Hilbert space (here
L
2
(T)) with respect to some (here it is the system of complex exponentials) complete
orthonormal basis. If we take the abstract view-point we are in the setting of a com-
pact Abelian group G, for which one can always find a family of characters

G forming
(automatically) a CONB for L
2
(G), hence we have an unconditional series expansion
f =

χ∈

G
¸f, χ¸χ , for f ∈ L
2
(G). (1)
Later on the condition of periodicity of functions could be given up, the Fourier series
theory was replaced by the technically much more challenging continuous Fourier trans-
form theory, but in both cases the question of inversion is rather delicate if understood
in a pointwise sense (see [6]). Later on A. Weil ( [61]) and others (Gelfand-theory)
pushed the theory of Fourier transform to its natural limits, as far as it concerned the
setting, namely the underlying LCA (locally compact Abelian groups). For the concrete
setting of Euclidean spaces that theory of tempered distributions developed by Lau-
rent Schwartzt ( [54]) provided the natural setting. Chronologically at the end of this
development (and there are books meanwhile on different versions of the fast Fourier
Transform) the FFT came in the picture, see [8].
But couldn’t we just try to revert this order? After all, the FFT routine realizes the
DFT (discrete Fourier transform), which is a simple and nice unitary mapping fromC
n
to
C
n
(up to the usual normalization factor of

n) and can be taught in any linear algebra
course. Looking at the matrix in a concrete way it turns out to be a Vandermonde matrix,
which allows to connect the properties of the finite Fourier transform with properties of
polynomials. In this way it is easy to explain why regular subsampling corresponds to
of the spectrum. Just for an illustration consider the polynomial p(t) = 1 + 2t
2
+ 3t
4
which can be interpreted as the lower order polynomial q(z) = 1 + 2z + 3z
2
, evaluated
at z = t
2
. But taking the squares of unit roots of order n is the same as running twice
through the unit roots of order n/2 (assuming that n is even).
7
But how can this help us to get from this finite setting to Fourier series and eventually
to the continuous Fourier transform? Can we say, that (for sufficiently large n at least)
the FFT allows is to compute a good approximation of
ˆ
f, at least for nice functions
f? What is the connection between the FFT of a sequence of regular samples of f and
corresponding samples (at which spacing?) of
ˆ
f ? What kind of setting would we need
to answer such a question?
Let us just mention here (because we will not be able to provide the details of the
answer in this note in sufficient detail): The distributional setting allows us to view finite
signals as periodic and discrete signals (if we use for a moment engineering terminology),
resp. as discrete periodic (hence unbounded, but translation bounded) measures, which
- properly described - will converge to a continuous limit in a very natural way!
2.2 Conceptual Harmonic Analysis
While Abstract Harmonic Analysis (AHA) was/is helpful for ignoring the technical dif-
ferences between the different settings, it is only establishing the analogy between differ-
ent groups. Already this has some advantages compared to the engineering approach.
Engineering books often make a distinction between continuous and discrete variables,
between one and several dimensions, between periodic and non-periodic signals, the AHA
perspective just asks for the identification of

G, given G. Depending on the context the
elements of the dual group are called characters, or pure frequencies or plane waves (see
e.g. [11]).
AHA also provides the general insight such as the fact that

G is discrete if and only
if G is compact, and vice versa. Also there is a natural identification of the dual group
of

G with G itself (formulated in the Pontrjagin van-Kampen duality).
The idea of Conceptual Harmonic Analysis (CHA) is a more integrative view-point.
In the Postmodern Area we have already all these tools, and we should try to put
things together in a more coherent way. In other words, we can do Fourier analysis in
the setting of Numerical Software Packages (many of us are using MATLAB, or other
packages), but we are also still interested in questions of continuous Fourier analysis.
We may thus ask questions like the following one: How can we compute the norm of the
Fourier transform of a given function in some function space, how can we approximately
identify the turning point or a local maximum of the Fourier transform of some explicitly
given Schwartz function f ∈ S(R
d
), preferably with a given precision, not depending
on the individual function, but only on qualitative information about it? Moreover, the
process should be computationally realizable, and questions of resources (such as time)
and memory requirements will enter the evaluation of any such an algorithm, which of
course will typically start with samples of the function taken over a regular grid, and
will somehow make use of the FFT. But how can it be done, what kind of new proofs
will be required, and under which conditions will the users be satisfied with the method
(e.g. because they can be assured that the performance is close to optimal).
Obviously plotting the FFT of a sequence of samples will not deliver such a result,
even if it might give (properly done) some indication of what one can expect. Note
that the result of the FFT is just finite sequence (maybe multi-indexed) of complex
numbers, while we are looking for a function, labeled by natural parameters, such as Hz,
to describe the acoustical frequency content of a musical signal.
8
This reasoning brings us to the idea, that the continuous limit is obtained in various
different ways, very much like Riemannian sums are approximating the Riemann inte-
gral. In other words, we are thinking of a family of approximating operations for our
target operation, which form a kind of Cauchy-net, because we clearly expect that two
sufficiently “fine” approximation are also close to each other in a suitable sense. But
which approach is computationally most effective, or most robust?
In summary, CHA (as we try to promote it) tries to emphasize the connections be-
tween the different settings in addition to the analogy already provided by the AHA
perspective. The current state of this approach still emphasizes the qualitative aspects.
We see some relevant results telling us that certain approximations work well asymp-
totically for a large class of functions. A typical representative of such a result is the
computation of
ˆ
f for f ∈ S
0
(R
d
), using only computations (FFTs) applied to (peri-
odized) function samples, as given in [40].
Of course we expect in the near future that aspects of approximation theory will have
to come into the picture, as well as of numerical analysis. Just think of the familiar
situation found in numerical integration: how densely should one sample in order to
guarantee that the Riemanian sum is sufficiently close to the integral, given some a priori
information about the smoothness of the function f which is to be integrated? To which
extent does the required sampling rate depend on this quality of f? And in addition:
aren’t there more efficient “numerical integration methods” (comparable to something
like the trapezoidal rule) that allow to achieve the same level of approximation at lower
computational complexity, at least for nice function, maybe not the most general ones
(i.e. the continuous functions, which are well suited for the Riemannian integral in its
full generality).
So at the end we see that the proposed setting requires to have an understanding
of questions reaching from approximation theory to numerical analysis and abstract
harmonic analysis, but in a more integrative sense than usual.
2.3 A Comparison with Number Systems
Of course the author has already tried to explain some of the proposed concepts to his
students, and in doing so a comparison with a familiar situation arising in the analysis
courses turned out to be helpful. More precisely we think about the use of the different
number systems, resp. the different fields which play a role for computations.
Let us therefore present these comparisons here in a nutshell. We also hope that the
perspective provided in this way will make it easier to readers less familiar with functional
analytic thinking (say a graduate engineering student) to appreciate the relevance of the
three layers proposed through our concept of Banach Gelfand Triples, which will play a
central role below.
Thus let us first recall that we all have learned about the “tower” of number systems.
First of all we have the rational numbers, and that it is the minimal field containing the
natural numbers. In other words, we need and get them just by carrying out the four
basic operations (addition etc.), starting from the natural number system N. In fact we
have just two important operations at hand, namely addition and multiplication. But
because all of them are supposed to be invertible (except for multiplication by zero), we
have in fact substraction (formally addition of the additive inverse element) and division
9
(multiplication with the multiplicative inverse element) at our disposition. Overall, there
are quite simple algebraic rules, and also compatibility between the two operations. In
particular, every child knows that the inverse to a/b is just b/a (if b ,= 0!).
We also have learned in our analysis courses that Q is incomplete, and that there
is no rational number x such that x
2
= 2. Hence there is/was a need to enlarge the
rational numbers. Although the p-adic numbers are a fascinating object (where the
completion is taken with respect to an alternative metric) we would like to remind
the reader hat this completion process leads again to a uniquely determined complete
field, if we measure distances in the Euclidean metric. This uniqueness implies that
one may work with different concrete models, knowing however that they are mutually
equivalent from the logical point of view (maybe not so much from the practical or
computational perspective). Of course the “representation” of real numbers as infinity
decimal expressions has a number of advantage, e.g. with respect to the ability of quickly
comparing the size of two numbers.
But this “enlargement process” (in the abstract setting done through equivalence
classes of Cauchy sequences) is in a way non-trivial and requires to carefully treat two
aspects. One has to properly embed the rationals into the reals, i.e. to provide an
injective mapping j : Q → R, i.e. to associate to each pair of rational numbers (p, q)
(with no common divisor) a well defined infinite decimal expression r ∈ R, and on the
other hand one has to extend the already existing multiplication of rational numbers the
new (and strictly larger) setting of real numbers. Of course we know in practice how to
do it. Given a pair of real numbers r
1
, r
2
we truncate their infinite decimal expression
at a certain decimal, thus obviously obtaining rational numbers, with denominator of
the form 10
k
for some k ∈ N, which consequently gives a rational and hence a real
number. Although this finite decimal expression is not just the truncation of r
1
r
2
it
is of course possible to verify that these products of truncated decimal expression is
convergent (for k → ∞) to some limit, i.e. more and more digits of the product can
be obtained exactly in this way, and we have a “natural multiplication” on R. Using
similar ideas one computes the multiplicative inverse, i.e. 1/r for r ∈ R.
This having said it is clear that numbers such as 1/

2 or 1/π
2
are well defined and
can be computed in many different ways, but the user does not have to care about the
actual realization of these operations, just the well-definedness of the object within R
counts and is sufficient in order to do interesting and correct mathematics.
At the end the impossibility of solving quadratic equations (such as x
2
+ 1 = 0)
suggests that one may need a further extension of the real number system, and in fact it
is a kind of miracle that the trick of “adjoining the complex unit” (or imaginary unit),
typically denoted by i or j (engineering), helps to overcome this problem.
But does this object really exist? Can we just write down a non-existent number?
After all, it has been called “imaginary unit”, indicating that it might not really exist,
and wishful thinking alone cannot solve a real problem!
As we know mathematicians have found a way to define the field of complex number
C through pairs of real numbers. So instead of a complex numbers z = a + ib we deal
with a pairs of real number (z is viewed as a point in the complex or Gaussian plane)
and define addition and multiplication properly, and verify that we have obtained a field
with respect to the addition and multiplication defined in the expected way.
Again one has to identify R with the subset of C consisting of elements of the form
10
(a, 0), a ∈ R, and that the new multiplication is (the only) compatible with the old one,
given for real numbers.
For our analogy we will use this comparison to indicate that “generalized functions”
(such as the Dirac measure) are not just ’vague objects’ but rather well defined objects,
one just has to be careful in manipulating them and follow clear rules, which are of-
ten motivated (if not uniquely determined in some way) by the behavior on “ordinary
functions”. So at the end complex numbers “exist” in the same way as ”bounded linear
functionals” exist in a very natural sense, allowing also computations with them, or the
application of linear operators (such as convolution operators or the Fourier transform,
which is originally defined as an integral transform on test functions, but this concrete
form of operation does not have to be meaningful in the more general setting, just as
the “inversion” of rationals is only trivial on the rationals, but not in the setting of real
numbers, where a more complicated procedure has to be applied).
So altogether we see a graded system of number systems, with a natural embedding
of one in to the other (larger one), so that whenever we are doing computations they can
be done either in the lowest possible level (e.g. using rationals) or at the highest level,
and the result will always be the same. In other words, there is a lot of consistency
within this graded system of fields, and ambiguities that might occur (from a logical
point of view) can be eliminated at the fundamental level, so that the user does not
have to care about such ambiguities at all, when doing practical work, i.e. computations
using any of these number systems.
We can learn from this example that a proper terminology and a well-defined set of
computational rules may simplify computations (in this case) very much. In fact, one
might argue that it is no surprise that mathematics was not of a high value within the
Roman empire, just because the system of Roman numerals is rather inadequate for
such a way of thinking.
2.4 Axiomatic Request from Conceptual Harmonic Analysis
We have just seen that the idea of Conceptual Harmonic Analysis (CHA) requires above
all a flexible setting which allows to describe important operations in a natural and easy
way.
Although a fine analysis of interesting (linear) operators requires to have the full
collection of suitable function spaces the reduction to a minimal set of spaces, namely
the so-called Banach Gelfand triple, will be enough. On the other hand, part of our
argumentation will be that the restriction to the Hilbert space setting alone would not
make sense. In fact, much of the problems in classical Fourier analysis are connected
with the concentration on the (too large spaces) L
2
(R
d
) and L
1
(R
d
).
The motivation comes partially from the success of the triple (S, L
2
, S

)(R
d
), consist-
ing of the Schwartz space of rapidly decreasing functions, the Hilbert space L
2
(R
d
) and
the dual space of “tempered distributions” S

(R
d
). It is also an example of a so-called
rigged Hilbert space, i.e. a Hilbert space endowed with some extra properties.
Since the topology on S(R
d
) (a Frechet space with respect to a suitable metric) is a
bit complicated, we suggest to look for a suitable Banach space of test functions. As a
consequence it is also possible to practically work with the dual space, either endowed
with the norm topology or (and we will see that this is an important extra structure) the
11
so-called w

-topology, the convergence in the strong topology (or pointwise convergence
of functionals).
So let us come to the axiomatic requests that one could ask for, supporting the
proposed concepts given later on.
First of all the AHA view-point suggests to ask for a functor associating to each
LCA group a triple of Banach spaces, consisting of a Banach space, densely embedded
into a Hilbert space (what else than L
2
(G)), and contained in the dual of the space,
a suitable space of tempered distributions, simply because we would like to be able
to properly describe the Fourier transform of continuous or discrete, periodic or non-
periodic functions in a unified way, and understand that unit vectors which are so useful
and simple to use in a (finite) discrete setting (e.g. in order to define the impulse response
of a translation invariant system) have to be replaced by Dirac’s Deltas, which are
definitely not treatable as “ordinary functions”. In fact, physicists have used for a long
time “point charges” or “point masses”, knowing of course that they are idealized limits
of true masses, which at least are normally not thought as smaller than the atoms.
Nevertheless such concepts are enormously useful in physics, and we would like to work
in a similar spirit.
The space of test functions also needs to have a rich internal and invariance struc-
ture. In particular it should be a regular algebra of continuous functions (see [52]) with
respect to pointwise multiplication, because only in this way we will be able to localize
distributions (elements of the dual space), hence define the support of a distribution.
Since we want to establish on quite general groups we cannot rely on the concept of
smoothness in the sense of infinity differentiability.
The space of test functions should of course be translation invariant, but also invari-
ant with respect to general automorphisms of the underlying group, e.g. rotations or
(isotropic or anisotropic) in the Euclidean case.
The analogy to the number system described in the previous section indicate that
the main features of this triple is the important operations (e.g. transformations can be
described as true integral transformation, without any problem with respect to technical
questions, inversion of transformation is equally simple, etc.) should be done smoothly.
The Hilbert space setting allows to express the fact the important mappings (such as the
Fourier transform) are unitary, even if integrals don’t exist in the proper sense for all the
elements, but the reduction to the dense subspace of test functions allows to overcome
such a technical question.
Finally, the dual space should be large enough to contain not only point measures
(equivalently: the space of test functions consists of continuous functions), or all the L
p
-
spaces (for 1 ≤ p ≤ ∞), but also e.g. the Dirac comb (often called the Shah-distribution)
.. =

n∈Z
d
δ
n
, because this is a central object in signal processing, e.g. in order to
describe the sampling process (as a multiplication operator by .. ). Poisson’s formula
can in fact be expressed then as the fact that T(..) = .. , which in turn implies that
sampling of the signal corresponds to a periodization of the spectrum supp(
ˆ
f) in the
frequency domain.
We do not claim that this collection of requirements uniquely determines the Banach
Gelfand Triple (S
0
, L
2
, S
0

)(G), based on the Segal algebra S
0
(G), which is well defined
according to the original paper [13], because the list of requirements has been formulated
too vaguely. However, in a very similar setting V. Losert has been able to prove the
12
uniqueness of S
0
(G) [44] under some very natural assumptions. Hence it is no surprise
that this “most natural Banach space” of test functions turns out to be very appropriate
in many different settings, not just in the context of Gabor analysis (where it is most
useful, e.g. as a reservoir of Gabor atoms).
3 The Banach Gelfand Triple (S
0
, L
2
, S
0
/
)
Because a general introduction to modulation spaces, and specifically the Segal algebra
S
0
(R
d
) can be found in other publications by the author (as well as in a number of talks,
downloadable from the NuHag Sever) let us just summarize the basic facts about this
specific space, which is well defined (and in a sense unique) for general LCA groups G
(see e.g. [13, 33] etc.).
In order to properly define this Banach space of continuous and integrable functions
on any LCA group we have to introduce the key players of time-frequency analysis first,
namely the time-frequency shifts and the STFT (Short-Time or Sliding Window Fourier
Transform).
T
x
f(t) = f(t −x) resp. M
ω
f(t) = e
2πiω·t
f(t) with x, ω, t ∈ R
d
These operators are intertwined by the Fourier transform
(T
x
f)= M
−x
ˆ
f , (M
ω
f)= T
ω
ˆ
f
Time-Frequency analysis (TF-analysis) starts with the Short-Time Fourier Transform:
V
g
f(λ) = ¸f, M
ω
T
t
g¸ = ¸f, π(λ)g¸ = ¸f, g
λ
¸, λ = (t, ω);
For any pair of functions f, g ∈ L
2
(R
d
) the STFT V
g
f is a bounded, continuous and
square integrable function over phase space, i.e. defined over R
d


R
d
. If |g|
2
= 1 then
the mapping f → V
g
f is even isometric, i.e. |V
g
f|
L
2
(R
d
×

R
d
)
= |f|
L
2
(R
d
)
. A function f
from L
2
(R
d
) belongs to (smaller) space S
0
(R
d
) if for some non-zero Schwartz function g
|f|
S
0
=

R
2d
[V
g
f(x, ω)[dxdω < ∞.
Different windows define the same space and equivalent norms. One has isometric invari-
ance

S
0
(R
d
), | |
S
0

under time-frequency shifts and under the Fourier transform. In
fact, one can show that S
0
(G) is the minimal Banach space of (say integrable) functions
with this property (see [13]). Consequently S
0
(G) is contained in any of the spaces L
p
(G)
(in the sense of continuous embedding) for 1 ≤ p ≤ ∞, while on the other hand its dual
space is large enough to contain all of them, in fact even containing any homogeneous
Banach space (B, | |
B
). This makes the pair

S
0
(G), | |
S
0

and

S
0

(G), | |
S
0


the
ideal setting to discuss Banach spaces of functions resp. distributions with such invari-
ance properties (see [4, 15] for discussions of some basic properties of such spaces).
Let us also not that a distribution σ ∈ S

(R
d
) belongs to the subspace S
0

(R
d
)
if and only if its spectrogram V
g
(σ) is a bounded over R
d


R
d
, with possible norm
|σ|
S
0
= |V
g
(σ)|

. The equally relevant w

-convergence in S
0

can be shown to be
equivalent to the very natural concept of uniform convergence over compact subsets of
R
d


R
d
. It is not difficult to show that pure frequencies resp. Dirac measures converge
(only) in this sense if their parameters converge.
13
4 How to make use of the BGT
The concept of Banach Gelfand Triples merges three possible view-points which are
playing a different role in different contexts. At the lower level typically descriptions can
be taken literal, i.e. integral transformations can be carried as as usual, often even just
by means of Riemannian integrals. The intermediate level allows to the preservation of
energy norms and scalar products, but sometimes limiting procedures are needed when
integral transforms are applied to general elements. Finally, the dual space is large
enough to contain objects such as Dirac measures (taking the role of unit vectors in the
discrete setting), even Dirac combs, or pure frequencies.
Let us again illustrate it through the example of the Fourier transform (also because
this is one of the central themes of Harmonic Analysis).
Here the roles of the different layers are clear. The space of test functions may be seen
as too small for several considerations
3
but it has the big advantage that almost all of the
problems normally associated with the use of the Fourier transform seem to disappear,
even if one is only willing to use absolutely convergent Riemannian integrals. In fact, the
Fourier inversion takes the expected form, it can be understood in the pointwise sense,
no sets of measure zero or measure theoretical arguments have to be involved, and even
Poisson’s formula is valid for all f ∈ S
0
(R
d
).
By way of “natural extension” the Fourier transform can be extended to the Hilbert
space L
2
(R
d
), becoming now a unitary isomorphism between L
2
(R
d
) and L
2
(

R
d
). In-
stead of L
1
∩ L
2
one simple considers another dense subspace, namely S
0
, and verifies
the isometry property of the FT on this dense subspace. L
2
(R
d
) is then simply the
completion of S
0
(R
d
) with respect to the L
2
-norm. Wether the Hilbert space consists of
“equivalence classes of measurable functions” (this is the stanadard model) or is just the
completion of S
0
(R
d
) with respect to the L
2
-norm, or a space of (regular) distributions
with certain integrability constraints does not matter for a wide range of application
areas. In contrast, the preservation of orthogonality is in fact an important practical
issue
4
.
Finally, the dual space S
0

(R
d
) is large enough to contain the pure frequencies (or
Dirac distributions), which are anyway always considered as the building blocks of the
Fourier transform, even if they do not belong to L
2
(R
d
).
4.1 The Segal Algebra

S
0
(G), | |
S
0

The Segal algebra

S
0
(G), | |
S
0

, introduced in [13] is not only a suitable space of test
functions, but also a very comprehensive reservoir of good functions as one would like
to use them for applications. In fact, if we wouldn’t have the well known concept of
measurable functions etc. one might think of the functions of S
0
(G) really as the “good
functions” which do not show any unexpected behavior. Although most of the properties
of

S
0
(G), | |
S
0

can be derived without great difficulty for the setting of G being an
LCA (locally compact Abelian) group, we concentrate here on the setting of R
d
, because
most readers will feel more comfortable in this setting. In particular, we can compare
3
Discontinuous functions do not belong S
0
(R
d
), in analogy to

2 being not a rational number.
4
Otherwise it would not be able to use digital radio, which communicate over different frequency
bands which are not overlapping, hence contain pairwise orthogonal signal spaces.
14
easily with the Schwartz space S(R
d
) of rapidly descreasing functions, which is much
more popular (and easier to handle) than the Schwartz-Bruhat space S(G).
Let us therefore list a number of cases within classical Fourier Analysis (and below
then in time-frequency analysis), where functions from the space S
0
(G) are “just the
right setting”. A series of papers by Franz Luef (see [45,46], [25]) also indicates that this
space is very useful in order to establish basic facts within non-commutative geometry.
4.2 Use of S
0
(R
d
) in Gabor Analysis
First of all the space S
0
(R
d
) of test functions is of course highly relevant, because it
allows to cover a number of aspects, in particular it provides a set of decent windows
(if one takes the analysis point of view) resp. of Gabor atoms when it comes to Gabor
synthesis. in fact, the following results described in all detail in [28], now just formulated
for the Hilbert space case, read as follows:
Theorem 1. Given g ∈ S
0
(R
d
) both the analysis operator C
g
: L
2
(R
d
) →
2
(Λ)
f → V
g
(f)(Λ)
and the synthesis operator R
g
:
2
(Λ) →L
2
(R
d
)
c →

λ∈Λ
c
λ
π(λ)g
are bounded, by constants which depend only on the S
0
- norm of g and geometric (cov-
ering) properties of the lattice Λ R
d


R
d
.
An important result in Gabor analysis is now the following:
Theorem 2. Assume that for some g ∈ S
0
(R
d
) Gabor family (π(λ)g)
λ∈Λ
is a Gabor
frame, then the canonical dual window ˜ g also belongs to S
0
(R
d
).
This is a rather deep theorem when taken in full generality, but not so difficult
to verify (using the Neumann series and Janssen’s representation of the Gabor frame
operator) whenever the given lattice is of good density (hence the adjoint lattice Λ

is
thin enough).
The combination of these two results actually provides a scenario for proving con-
tinuous dependence of dual frames on both the atom g ∈ S
0
(R
d
) and the lattice. Note
that the continuous dependence of TF-shifts when applied to an individual f ∈ LtRd we
also find that R
g
(c) depends continuously on Λ (with the natural convergence of lattices
expressed by their generating matrices), but this is not a perturbation in the operator
norm sense!
Nevertheless we have the following robustness result [21]:
Theorem 3. Assume that (g, Λ) with g ∈ S
0
(R
d
) generates a Gabor frame or a Gaborian
Riesz basic sequence. Then for all g
1
, close enough to g in the S
0
-sense, and Λ
1
close
enough to Λ the corresponding family derived from (g
1
, Λ
1
) is of the same type. Moreover,
the canonical dual generators depend continuously (in the S
0
-sense) on both variables.
15
This result is not only of theoretical interest, but allows also to approximate a general
continuous problem by a corresponding “rational problem” which in turn can be well
approximated by a finite problem, susceptible to good approximation by the finite (and
computable) setting. How such an argument can be used to approximately compute
the dual window for a Gaussian window and a general lattice in R
2
is described in [10]
(together with alternative methods).
4.3 Fourier Inversion and Summability
Although at first sight

L
1
(G), | |
1

appears as the natural domain for the Fourier
transform (if/since it is first of all viewed as an integral transform) there is some hidden
asymmetry when it comes to the inversion. In fact, as it is well known
ˆ
f / ∈ L
1
(
ˆ
G) for
any non-zero, discontinuous function, and thus recovery of f from
ˆ
f is not simply by
applying the Fourier inversion formula. This has lead to the introduction of summability
kernels. A parameterized family of functions (h
α
) can be used to recover f from
ˆ
f if it is
a bounded family in the Fourier algebra (resp. sup
α
| Th
α
|
1
< ∞ ) and [h
α
(s) −1[ → 0
with α, uniformly over compact sets, with h
α
∈ L
1
(
ˆ
G).
A large reservoir of such summability kernels are obtained by taking a “decent func-
tion” and simply applying dilations to it, i.e. to choose h
α
= D
α
h, with D
ρ
h(z) = h(ρz).
Taking then the limit α → 0 provides the proper setting, covering a rich variety of clas-
sical cases.
It is easy to argue that is enough to assume that h ∈ S
0
(R
d
) with h(0) = 1. In
fact, in this case h = ˆ g for some g ∈ S
0
(R
d
) with

R
d
g(x)dx = 1. Since the simple
dilation operator on Fourier transform corresponds to L
1
-norm preserving stretching,
i.e. St
ρ
g(x) = ρ
−d
g(x/ρ), we find that hf ∈ TL
1
(R
d
) S
0
(R
d
) ⊂ S
0
(R
d
) ⊂ L
1
(R
d
) and
on the other hand f = lim
α→0
St
α
g ∗ f in the norm for any homogeneous Banach space,
so for example in L
p
(R
d
), for 1 ≤ p < ∞ or f ∈ C
0
(R
d
).
4.4 Poisson’s Formula
Poisson’s formula is a cornerstone within both abstract and applied harmonic analysis.
In the finite setting it is an easy consequence of the fact, that except for N = 1 all
the unit-roots of order N add up to zero, verified by computing finite geometric sums,
involving units roots of order N. Using the principle that pointwise multiplication of
signals (e.g. a sum of Diracs) corresponds to convolution with its Fourier transform if
such a Dirac comb. Poisson’s formula tells us that this is again a Dirac comb, hence
sampling in the time or image domain is equivalently described by periodization of its
Fourier spectrum.
The classical Poisson formula (given here in a normalized version in the setting of
R
d
), claims that under “suitable conditions on f” one has

k∈Z
d
f(k) =

n∈Z
d
ˆ
f(n).
Unfortunately (as already pointed out in Katznelson’s book, see [41]), it is not enough
to assume integrability of f and
ˆ
f, nor is the absolute convergence of the sum on both
sides.
16
Kahane and Lemarie [39] were able to give pairs of weighted L
p
-conditions which still
allow to come up with counter examples to Poisson’s formula (in the pointwise sense,
as stated above). It was then K. Gr¨ ochenig (see [32]) who was able to show that in
all the cases where the combined conditions on the function and its Fourier transform
are strong enough to avoid this unpleasant situation, i.e. to guarantee the validity of
Poisson’s formula one has already an embedding into S
0
(R
d
). On the other hand for
f ∈ S
0
(R
d
) the validity of Poisson’s formula is obvious (at least once the characterization
of S
0
(R
d
) using atomic decompositions is known, see [13]).
Thus altogether it is not wrong to argue that S
0
(G) is the largest universally defined
Banach space of continuous functions such that Poisson’s formula is valid for general
co-compact lattices Λ G (because in this case Λ

is again a discrete lattice in

G).
This claim not only valid for the ordinary Fourier transform, but also for the so-called
symplectic Fourier transform, where the validity of the symplectic Poisson formula can
be used to prove the fundamental identity of Gabor analysis in its most general form
(for details see [25]):
4.5 Fourier Transform’s of Unbounded Measures
In the work of L. Argabright and J. Gil de Lamadrid [2] a theory of (potentially un-
bounded) measures is developed having the property that in some sense their Fourier
transform is still meaningful and can be understood as a (regular Borel) measure on the
dual group. With their approach it was not possible to clarify the asymmetry of the
setting, arising from implicit boundedness assumptions. In fact, transformable measures
in their sense have to be translation bounded while there transform just need to be lo-
cally a bounded measures, without global restraints. Using tools involving S
0

(R
d
) one
could rephrase some of their results by saying, that symmetry is established by looking
at those (still possible unbounded) measures which are translation bounded (i.e. belong
to the Wiener amalgam space W(M,

)) together with their (distributional) Fourier
transforms. Again, Poisson’s formula, i.e. in the most general setting, the Fourier pairing
of the ..
Λ
(the Dirac comb over a lattice Λ R
d
) and ..
Λ
⊥, the Dirac comb over the
orthogonal lattice Λ



G provides a good example of such a situation.
4.6 Multipliers
In the discussion of multipliers between different translation invariant spaces the book
of Larsen [42] a large number of multiplier spaces is described, sometimes using rather
difficult concepts, such as the concept of quasi-measure as introduced by G. Gaudry
[30]. It is also shown by Gaudry that not only the translation invariant system can
be represented by a convolution with a quasi- measure, but also transfer function is a
quasi-measure. This looks nice at first side (engineers would say that there is a well-
defined impulse response, and its Fourier transform, also a quasi-measure, is the transfer
“distribution”), but unfortunately general quasi-measures (although behaving decently,
namely like pseudo-measures locally) may not have a Fourier transform, not even in the
sense of tempered distributions.
Within the context of the S
0
-BGT the situation is quite simply described. Every
operator from L
p
to L
q
(with p < ∞) commuting with translations can also be viewed
17
(via restriction to a dense subspace) as an operator T from S
0
(G) to S
0

(G) (commuting
with translation). Any such operator maps S
0
(G) in fact already into C
b
(G) ⊂ S
0

(G),
and consequently it is not difficult to find out that the linear functional σ ∈ S
0

(G),
defined by σ(f) = T(f

)(0)
5
is the appropriate convolution kernel. Obviously ˆ σ ∈
S
0

(

G), and since S
0

is always continuously embedded into the quasi-measures (how
complicated they may be defined) the above mentioned characterization is quite clear.
4.7 The dual S
0
/
(G) of the Segal Algebra S
0
(G)
The theory of tempered distributions, using S(R
d
) and S

(R
d
) by L. Schwartz has been
tailored to the needs of Partial Differential Equations, with the consequence that a family
of unbounded operators (including differentiation and multiplication with polynomials,
as well as the Fourier transform) have to be continuous, and consequently the topology
of S(R
d
) is all but easy to describe. It is even less easy to convince engineers the make
use of it, and it is in fact not necessary to have distributions of this generality for the
majority of application problems. However a proper handling of the δ−distribution or of
Dirac combs to describe the sampling and/or periodization procedures is very important
for a good understanding. Fortunately S
0

(R
d
) is large enough for this purpose, but still
easy to handle technically. The only new aspect that will be emphasized is the use of
two topologies on the space, namely the norm convergence, but even more importantly
the w

-convergence. Although this notation is less familiar among engineers one should
point out that the simple fact that Riemannian sums converge to the integral is an
instance of this type of convergence, indicating that the concept is not at all a “very
complicated one”.
5 The Role of Computational Harmonic Analysis
Distribution theory is not only an important tool for real or harmonic analysis. It is also
needed whenever one wants to describe the approximation of a continuous scenario by
discrete, resp. periodic or even finite dimensional models. Since the functions from such
models can be seen as discrete, periodic signals, one cannot describe their convergence
to say some L
2
(R
d
)-function using the L
2
(R
d
)-norm, simply because those periodic and
discrete signals do not belong to L
2
(R
d
), not even locally.
It is the concept of w

-convergence which help us to give a proper mathematical
description of the situation. So far this can be done only at a qualitative (but quite
general) level, a good example being the paper [40] (approximate computation of the
dual Gabor atom resp. the Fourier transform using FFT based methods) or [19, 27], but
a more quantitative approach in the spirit of approximation theory has still to be done.
5.1 Verification by Computation
So far it appears to me that most of the time theoretical results have been first derived,
and afterwards somebody (else) was trying to find out how the insight can be transferred
to a computational setting, how the concrete numbers can be computed, and so on.
5
We use the standard convention f

(z) := f(−z) here.
18
The explorative way of using MATLAB experiments to find out what the “continuous
truth” has been exercised by a relatively small minority of researchers working on Fourier
analysis. There is too much separation between “true numerical analyst” and people
working in abstract harmonic analysis. But fortunately it is becoming more and more
popular among younger colleagues to carry out computational work or simulations in
order to gain insight. Nevertheless we should not only improve the existing software and
make use of more and more powerful computer systems, but the community should also
systematically develop reliable software packages for specific subject areas (which then
should be integrated to more comprehensive, easy to use packages in the future).
For me personally sometimes the motivation was, boldly formulated: Why should I
look for a proof if the computer indicates to me that there might be counter-examples,
coming up occasionally during a series of simulations. Or in the opposite direction: I
observe that something is happening regularly (e.g. a certain preconditioner works well).
Then immediately the question arises: is this due to the setting I have chosen, or is it
a more general phenomenon. Whatever the answer is: if the observations are occurring
in a consistent way, also with slightly different settings, can one distill a theoretical
statement out of such observations.
Another, not so wide-spread idea would be the computational verification of esti-
mates. If we can compute norms or other numbers arising during a series of analytic
estimates, all of which can be computed (at least approximately). Shouldn’t we check
to which extent these estimates are valid and close to optimal? Even if they are optimal
the worst case might occur in very few rare cases in an application setting, so it might
be worth to find out more about this. Concepts from compressed sensing (claiming that
certain random experiments will be successful with overwhelming probability if certain
assumptions are met) are very much in this direction and should be further pursued,
also in related contexts.
Of course the suggested use of new tools will raise new and interesting questions of
the following type: What is the most efficient way to verify the validity of inequalities? If
tests should be carried out on a variety of cases via simulations maybe a rough estimate,
say up to some constant applied to a huge number of cases might be more informative
as compared to an almost perfect estimate in a single case. Can we proof analytically
that our tests are providing some guarantees, or are they just heuristic tools? How can
one minimize the overall computational effort while maximizing the insight gained by
such computations? Are there general strategies which can be recommended, and how
should we teach the necessary perform such tasks, not just at an individual level.
5.2 User Guides and Consumer Reports
Let me mention some experiences related to the recent EU project UnlocX, where essen-
tially the following question was addressed by our team: What is a good Gabor system,
i.e. a pair of an atom g and a lattice Λ in the TF-plane such that the corresponding
family (π(λ)g)
λ∈Λ
is a Gabor frame.
Just as a first step we have studied the finite case, i.e. signals of finite length (resp.
functions on Z
N
resp. discrete periodic functions). First we had to be able to compute
the generator of the canonical and tight Gabor frames for general lattices, including the
non-separable ones. This in turn raised the questions whether we can generate all of
19
them, maybe even produce a given list of such lattices Λ at a given redundancy (there
are only finitely many possible rational redundancies for a given integer N, and only
those not too high are typically of interest for applications), and of course this should
enable to user to run an exhaustive search.
Once being able to do this we had to come up with a variety of quality criteria,
among them the condition number of the frame operator, the S
0
-norm of the dual atom
or geometric properties of the considered lattices using maximal pavements and minimal
coverings. Only systematic and exhaustive computations of various (other) figures of
merit then allowed us to identify the most relevant criteria and in fact to establish (in
this case) the ranking derived from each of them turned out to be more or less equivalent.
As a further step one then has to find out which version of the criterion is the most
efficient one, and how to select from the full variety of possible lattice the most interesting
or most suitable ones for whatever specific are application (and corresponding figures of
merit).
In fact, at the end we are not far from a consumer’s report, which gives a customer
(in our case maybe an engineer) advice of which system of functions might be most
suitable for her/his application, or which combinations of Gabor atoms with TF-lattices
Λ can be recommended resp. might not be as good as expected according to such an
analysis.
An interesting point is the fact, that the justification for the validity of a hopefully
unbiased recommendation requires itself quite a bit of sophisticated analysis, which is
interesting in itself, but has found little attention so far.
5.3 Fourier Transform in Practice
The FFT (resp. DFT) is understood as the “natural way of doing a (continuous) Fourier
transform”, because it looks so similar, and after all a computer can only handle finite
data. This is a typical engineering argument, which is of course true, but perhaps
not convincing for people in numerical analysis or those who have to take care of the
implementation of some algorithm on a DSP looking for an “efficient implementation”
of some idea coming up in constructive approximation theory.
For many of those working regularly with the FFT the mathematical viewpoint
(that CMH would advocate), namely to view what is done in the finite setting as an
approximation of the continuous problem by corresponding problem over Z
N
, or to treat
it as a question of approximating a distributional entity in the w

-sense by a distribution
which can be thought as living on Z
N
(i.e. a discrete and periodic distribution) would
be seen as an overkill in terms of complicated concepts.
On the other hand practical books, especially quite recent ones, such as [58] or [?]
indicate to their readers that one has be to quite careful in the interpretation of e.g.
the values of the output of the FFT (which coordinate corresponds to which frequency,
etc.), and a plethora of “tricks” has to be learned by the user, most of which would be
relatively easily explained through the CMH context.
In fact, there was perhaps not enough interest from both sides to build and enforce
this bridge (meaning engineers and mathematicians), but I see a formidable task here,
to improve the understanding and the teaching of these subjects. An interesting recent
source is also [1].
20
6 The relevance of w

-convergence
There are many situations in Fourier analysis where heuristic arguments are used to
describe the transition from one setting of Fourier analysis to another. For example,
one often finds the formulation (used also by engineers): On a computer we can only
deal with finite sequences and therefore instead of computing the Fourier transform of
f ∈ L
1
(R
d
) we have to apply the FFT to a sequence of samples. Another typical case
is the approximation of the Fourier transform by thinking of the integral transform as
of the limit of Fourier series expansions for the given function f ∈ L
1
(R
d
), viewed as
a p−periodic function (obviously defined, at least if f has compact support), by taking
the limit p → ∞. We plan to give a variety of such examples in the rest of this section.
6.1 Fourier Integrals as Limits of Fourier Series
One of the spots where most presentations of the Fourier transform have a hard time to
explain in a more than purely heuristic way why the Fourier transform (and its inverse)
take their classical shape is the way how the Fourier transform can be understood as the
limiting case of the classical theory of Fourier expansions of periodic functions, just by
letting the length of the period go to infinity.
First let us view the theory of Fourier series in our distributional setting. This is
not a new idea, but just a refreshment of known ideas from the context of tempered
distributions, cast into the more simple setting of S
0

(R) (again things are valid for
general LCA groups). Thus we have to reverse an earlier learning process, where at the
beginning of a course on Fourier series we are told to view periodic functions (assuming
furthermore that the period is known!) as functions on the torus. Now we take to simple
classical viewpoint of considered a periodic function as a locally integrable function with
some periodicity condition, in our case T
p
f = f for some p > 0 (resp. T
λ
f = f ∀λ ∈ pZ).
Using now a partition of unity, for example a BUPU (bounded uniform partition
of unity) of the form (T
λ
(ψ)), hence satisfying

λ∈Λ
T
λ
ψ ≡ 1, with ψ some compactly
supported function and Λ = pZ. The set of translates of cubic B-splines (appropriately
dilated) are a good example for this situation. Hence h := f ψ has the property that
f =

λ∈Λ
T
λ
h = ..
Λ
∗h, with Λ = pZ. Then (up to some constant) one has
ˆ
f = T(h ∗ ..
p
) =
ˆ
h (1/p) ..
1/p
,
or in other words
ˆ
f =
1
p

n∈Z
ˆ
h(n)δ
1/p
or equivalently
f =
1
p

n∈Z
ˆ
h(n/p)χ
n/p
,
telling us that in turn the (p−periodic) is (at least in some sense) an infinite sum of pure
frequencies from the lattice Z/p, which is getting more and more dense (within R) as
p → ∞. The correct way of verifying this convergence is again in the w

-setting. Note
that 1/p ..
p
is convergent to the constant 1 (resp. to the Haar measure on R
d
) in the
w

-sense.
21
6.2 Generalized Stochastic Processes
The last two decades have shown that the BGT (S
0
, L
2
, S
0

) is well suited in the modeling
of slowly varying (resp. underspread) channels, as they arise in mobile communication.
We do not elaborate on this, but would like to mention that there are new publications
coming up on this direction, where the setting (originally developed in the late 80’s) of
generalized stochastic processes is taken up again. The idea is relatively simple, of one
takes a completely abstract view-point, and takes the space of random variables with
expectation value zero and finite second order moments as an (more or less abstract)
Hilbert spaces. Then one can identify a generalized stochastic process over an arbitrary
LCA group G with a linear operator from S
0
(G) to this Hilbert space 1. This approach
to generalized stochastic processes have been developed in detail in the PhD thesis of
W. H¨ormann [37] or in the technical report [20] (see also [60] for related results).
In this setting it is quite easy to show that the auto-correlation σ
ρ
if such a process
ρ belongs to S
0

(G G), that there is a spectral process ˆ σ
ρ
, whose auto-correlation is
just the distributional Fourier transform of σ
ρ
. Stationarity can be expressed naturally
in this context, and many of the good properties of the space S
0
(G) (over general LCA
groups) allow to provide proofs of essential facts without relying on specific techniques
of vector-valued integration, as it is required in the standard approach.
We do not go into details here, but would like to point out that our knowledge of
the temperature distribution (given as an example above) is probably modeled better in
this setting, since in fact one does not have exact (but rather probabilistic) information
about the actual temperature, and also not at a given point in the 4D scenario, but
rather average values (e.g. because every temperature measurement needs some time
and a non-trivial volume element).
We leave it to the readers to think about more complicated situations and the po-
tential of such a view-point, perhaps also with the possibility of proper simulation of
such generalized stochastic processes on the computer.
7 Summary
The purpose of this note was to suggest a fresh look on (abstract and applied) Harmonic
Analysis. We suggest to view HA as a natural subfield of functional analysis, dealing with
Banach spaces of functions and distributions, with a variety of group actions on (and
between them), above all the Fourier transform, time-frequency shifts resp. dilations in
the context of wavelet theory.
This view is in remarkable contrast to the by now classical view-point of Modern
Harmonic Analysis
6
which emphasizes the role of L
p
-spaces, maximal functions, and
almost everywhere concepts. But if one looks into the (really) applied literature the
L
p
-spaces play a surprising modest role, except for the Sobolev spaces derived from L
2
,
which are not just Banach spaces but even Hilbert spaces with a suitable scalar product,
and of course L
1
-spaces which appear as natural domain for the Fourier transform (taken
as integral transform) or convolution of functions.
6
Slightly different from the concept of “Hard Analysis”.
22
Abstract Harmonic Analysis in turn emphasizes the pure analogy between the dif-
ferent settings, whether one deals with the Fourier transform over the cyclic group Z
N
of order N in order to explain the properties of the FFT and the connection to cyclic
convolution, or whether one is dealing with the Fourier transform over the Euclidian
space R
d
or over the p-adic numbers.
The new approach outlined below is giving a higher priority to the distributional
setting, via the use of Banach Gelfand Triples (resp. rigged Hilbert spaces), originally
motivated by studies in time-frequency analysis. Meanwhile the setting has found some
recognition as a versatile tool in a much more general setting, including the study of
pseudo-differential operators. The distributional setting also provides the proper setting
for the description of otherwise vague concepts, allowing a proper description of the con-
nection between the different settings (e.g. continuous and non-periodic versus periodic
and discrete, i.e. finite), using the classical notion of w

-convergence. At the end we
hope that the reader may not find it surprising that the approach gives a handle on
classical procedures like the use of Fourier summability kernels (as used for the Fourier
inversion theorem), or sampling resp. periodization processes.
8 Final Remarks
I hope that this essay will spur interest in reading other, certainly more elaborate essays
on harmonic analysis, such as those of G. Mackey [47, 48] or J.M. Ash ( [3]) and rethink
the goals of harmonic analysis.
Last but not least let me suggest at least to young researchers in the field to keep the
eyes open for application areas where still a lot of mathematical tools based on Fourier
Analysis needs to be developed resp. where applied scientist (e.g. optics, astronomy,
signal processing, ...) perform computations or simulations involving Fourier methods
on a very intuitive and practical way, without reliable and detailed theoretical analysis
behind. It is in such territories where new mathematics is needed and new problems
and concepts have to be developed. Fourier analysis has a long and remarkable history
to show. It has not reached the end of it’s life span, and done properly, it will also a
bright future!
8.1 Acknowledgements
The author would like to thank Franz Luef for many discussions of the general concepts
described in this note during the last years. He also provided occasionally relevant
bibliographical hints.
Several joint papers, with my coauthors W. Kozek ( [23,24]), G. Zimmermann ( [28]),
N. Kaiblinger ( [21, 22]), F. Luef ( [25, 26]) and E. Cordero ( [9]) describe concrete parts
of the program outlined in this note.
23
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