THE Banach Gelfand Triple and Its Role in Classical Fourier Analysis and Operator Theory

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INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm.

Numerical Harmonic Analysis Group

THE Banach Gelfand Triple and its role in Classical Fourier Analysis and Operator Theory

Hans G. Feichtinger, Univ. Vienna [email protected] www.nuhag.eu TALKS at www.nuhag.eu/talks

Internet Presentation f. Southern Federal Unverisity Regional Mathematical Center, 25.06.2020 Supervisors: A.Karapetyants, V. Kravchenko Orig.: Invited Talk, IWOTA 2019, 23.07.2019 THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Hans G. Feichtinger, Univ. Vienna [email protected] www.nuhag.eu TALKS at www.nuhag.eu/talks INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. A Zoo of Banach Spaces for Fourier Analysis

Psychological Viewpoint: We are coming close to 200 years of the Fourier transform. It is an important and indispensable tool for many areas of mathematica analysis, function spaces, PDE and other areas. It has been crucial in the creation ofnew tools, such asLebesgue integration, Functional Analysis, Hilbert spaces, and more recentlyframes (for Gabor expansions).

Analogies: The DFT (realized as FFT) is a simple orthogonal n (better unitary) change of bases, linear mappings on C are just d matrices, but when it comes to continuous variables (FT over R ) only physicists call (δx )x∈R a “continuous basis” for (generalized) d functions on R .

THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Hans G. Feichtinger INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. NEW Abstract (short version), Rostov-on-DonII Technically Speaking: Engineers work with Dirac impulses, Dirac combs, and more recently to some extent the theory of tempered 0 d distributions (S (R )) by Laurent Schwartz is taken into account. “Non-existent” integrals and identities such as Z ∞ 2πits e ds = δ0(t) −∞ are used to “derive” the Fourier Inversion Theorem and other mysterious manipulation give engineering students the impression that Fourier Analysis is a mystic game with fancy symbols which is only understood correctly by mathematicians. On the other hand they only care about applications, and mathematicians are often not helpful (e.g. because they are viewed as too pedantic).

THESIS of my TALK

S L2 S0 d The Banach Gelfand Triple ( 0, , 0)(R ) (which arose in the context of Time-Frequency Analysis) isa simple and useful tool, both for the derivation of mathematically valid theorems AND for teaching relevant concepts to engineers and physicists (and of course mathematicians, interested in applications!). In this context the basic terms of an introductory course on Linear System’s Theory can be explained properly: Translation invariant systems viewed as linear operators, which can be described as convolution operator by some impulse response, whose Fourier transform is well deﬁned (and is called transfer function), and S d S0 d there is a kernel theorem: Operators T : 0(R ) → 0(R ) S0 2d have a “matrix representation” using some σ ∈ 0(R ).

It is the purpose of this presentation to explain certain aspects of Classical Fourier Analysis from the point of view of distribution theory. The setting of the so-called Banach Gelfand Triple S L2 S0 d ( 0, , 0)(R ) starts from a particular Segal algebra S d 0(R ), k · kS0 of continuous and Riemann integrable functions. It is Fourier invariant and thus an extended Fourier transform can 0 d be deﬁned for (S ( ), k · kS0 ) the space of so-called mild 0 R 0 p d distributions. Any of the L -spaces contains S0(R ) and is S0 d embedded into 0(R ), for p ∈ [1, ∞]. We will show how this setting of Banach Gelfand triples resp. rigged Hilbert spaces allows to provide a conceptual appealing approach to most classical parts of Fourier analysis. In contrast to the Schwartz theory of tempered distributions it is expected that the mathematical tools can be also explained in more detail to engineers and physicists. THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Hans G. Feichtinger INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. Our Aim: Popularizing Banach Gelfand Triples

According to the title I have to first explain the term Banach Gelfand Triples, with the specific emphasis on the BGTr S L2 S0 d S d ( 0, , 0)(R ), arising from the Segal algebra 0(R ), k · kS0 (as a space of test-functions), alias the modulation spaces 1 d 2 d ∞ d (M (R ), k · kM1 ), L (R ), k · k2 and (M (R ), k · kM∞ ). Hence I will describe them, provide a selection of different characterizations (there are many of them!) and properties.

Finally I will come to the main part, namely applications or use of this (!natural) concept in the framework of classical analysis.

Compared to other talks I leave out the main application area of this BGTr, namely so-called Gabor Analysis or Time-Frequency Analysis (TFA). Speciﬁc aspects of this have been described in my talks in the Gabor Analysis section of this conference.

S In fact, the Segal algebra 0(G), k · kS0 , which is well deﬁned on LCA (locally compact groups) has been introduced already 1979, at a winter-school in Vienna, organized by H. Reiter (my advisor). S L2 S0 This makes the BGTr ( 0, , 0)(G) useful for applications in Abstract Harmonic Analysis.

New tools often go through the following development steps:

1 A new tool is developed in order to solve a concrete problem; 2 Once it is observed that the new tool can be applied in other contexts the “trick” becomes a method, and receives a name; 3 Subsequent analysis of the methods specifies the ingredients and conditions of applications; optimization; 4 Exploration of the maximal range of applications; 5 The limitations of the tool are discovered, and so on... 6 Change of view/problems creates the need for new tools! 7 One may find that for various applications a simplified version suffices.

THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Hans G. Feichtinger 1 2 1 Abbildung: The diagrams showing L , L , C0, FL etc. INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. The Universe of Banach Spaces of Tempered Distributions

THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Abbildung: Our GOAL:Hans Simpliﬁcation: G. Feichtinger COMPARISON with Q ⊂ R ⊂ C INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. frametitle: Bibliography for Banach Gelfand TriplesI

E. Cordero, H. G. Feichtinger, and F. Luef. Banach Gelfand triples for Gabor analysis. In Pseudo-diﬀerential Operators, volume 1949 of Lecture Notes in Mathematics, pages 1–33. Springer, Berlin, 2008. H. G. Feichtinger and M. S. Jakobsen. Distribution theory by Riemann integrals. Mathematical Modelling, Optimization, Analytic and Numerical Solutions, pages 33–76, 2020. H. G. Feichtinger. A sequential approach to mild distributions. Axioms, 9(1):1–25, 2020. H. G. Feichtinger. Classical Fourier Analysis via mild distributions. MESA, Non-linear Studies, 26(4):783–804, 2019. H. G. Feichtinger. Banach Gelfand Triples and some Applications in Harmonic Analysis. In J. Feuto and M. Essoh, editors, Proc. Conf. Harmonic Analysis (Abidjan, May 2018), pages 1–21, 2018.

There are many Segal algebras in the sense of H. Reiter. 1 By deﬁnition they are dense Banach spaces within L (G), k · k1 with isometric and strongly continuous translation, hence (Banach) 1 d ideal within L (G), k · k1 , for any LC group (here G = R ). The intersection of all those Segal algebras is NOT a Segal algebra itself, but if one adds additional pointwise multiplication properties one can ﬁnd a smallest element in such a family.

d For the simple choice A = C0(R ), k · k∞ the Wiener algebra 1 d (which I use to denote by) W (C0, ` )(R ), k · kW is the L1 d corresponding Segal algebra, while for F (R ), k · kFL1 it is the S mentioned Segal algebra 0(G), k · kS0 . So let us consider a FEW out of the MANY diﬀerent characterizations of this Banach spaces (of continuous and d integrable) functions on G = R .

Short Bibliography of related to Segal Algebras H. Reiter. Classical Harmonic Analysis and Locally Compact Groups. Clarendon Press, Oxford, 1968. H. J. Reiter. L1-algebras and Segal Algebras. Springer, Berlin, Heidelberg, New York, 1971. H. G. Feichtinger. A characterization of Wiener’s algebra on locally compact groups. Archiv d. Math., 29:136–140, 1977. H. G. Feichtinger. On a new Segal algebra. Monatsh. Math., 92:269–289, 1981. H. Reiter and J. D. Stegeman. Classical Harmonic Analysis and Locally Compact Groups. 2nd ed. Clarendon Press, Oxford, 2000. M. S. Jakobsen. On a (no longer) New Segal Algebra: A Review of the Feichtinger Algebra. J. Fourier Anal. Appl., 24(6):1579–1660, 2018.

What we need in order to give the ﬁrst, simple deﬁnitions (for the d case of G = R ) are the following ingredients: Time-shifts and Frequency shifts (II)(modulations)

Tx f (t) = f (t − x) d and x, ω, t ∈ R 2πiω·t Mωf (t) = e f (t) . Behavior under Fourier transform (modulation goes to translation) ˆ ˆ (Tx f )b= M−x f (Mωf )b= Tωf

The Short-Time Fourier Transform

Vg f (λ) = hf , MωTt gi = hf ,π (λ)gi = hf , gλi, λ = (t, ω);

Abbildung: GauRotMod1.jpg THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Hans G. Feichtinger INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. Alternative Function Spaces

d There are many situations where the Segal algebra S0(R ) is the most appropriate function spaces in the context of Gabor Analysis. We will ﬁrst describe the space shortly (using a few of its characterizations) and then list a few examples where and why it is a convenient and frequently used tool in the context of Gabor Analysis. d 1 The (meanwhile) classical way to introduce S0(R ) is making use 2 d of the Short-Time Fourier transform of a signal f ∈ L (R ) with 2 d respect to a window g ∈ L (R ) is deﬁned by d Vg (f )(t, ω) = hf , MωTt gi, t, ω ∈ R . (1)

Fixing any non-zero Schwartz function, typically g = g0, with −πt2 g0(t) = e , one sets: d 2 1 2d S0(R ) := {f ∈ L (R) | Vg (f ) ∈ L (R )}. (2) 1This so-called Segal algebra can be deﬁned over general LCA groups. THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Hans G. Feichtinger INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. M1 S d Basic properties of = 0(R )

Lemma

d Let f ∈ S0(R ), then the following holds: d d d (1) π(u, η)f ∈ S0(R ) for (u, η) ∈ R × Rb , and

kπ(u, η)f kS0 = kf kS0 . ˆ S d ˆ (2) f ∈ 0(R ), and kf kS0 = kf kS0 . S d In fact, 0(R ), k · kS0 is the smallest non-trivial Banach space with this property, and therefore contained in any of the Lp-spaces (and their Fourier images).

The Moyal formula implies immediately the following inversion formula “in the weak sense” Z 2 f = Vg (f )(λ)π(λ)gdλ, f ∈ L (R). (3) Rd ×Rbd It follows from Moyal’s formula (energy preservation):

L2 kVg (f )k 2 d d = kgk2kf k2, f , g ∈ . (4) L (R ×Rb ) This setting is well known under the name of coherent frames when g = g0, the Gauss function. Its range is the Fock space. d But for g ∈ S0(R ) one has (according F. Weiss) even norm 2 d convergence of Riemannian sums, for any f ∈ L (R ).

Short Bibliography of Wiener Amalgams H. G. Feichtinger. A characterization of Wiener’s algebra on locally compact groups. Archiv d. Math., 29:136–140, 1977. F. Holland. Harmonic analysis on amalgams of Lp and `q. J. Lond. Math. Soc., 10:295–305, 1975. R. C. Busby and H. A. Smith. Product-convolution operators and mixed-norm spaces. Trans. Amer. Math. Soc., 263:309–341, 1981. H. G. Feichtinger. Banach convolution algebras of Wiener type. In Proc. Conf. on Functions, Series, Operators, Budapest 1980, volume 35 of Colloq. Math. Soc. Janos Bolyai, pages 509–524. North-Holland, Amsterdam, Eds. B. Sz.-Nagy and J. Szabados. edition, 1983. J. J. F. Fournier and J. Stewart. Amalgams of Lp and `q. Bull. Amer. Math. Soc., New Ser., 13:1–21, 1985.

It is the purpose of this talk to highlight a few topics arising from Gabor Analysis, which are also of great use also for other branches of functional analysis and application areas such as communication theory or (quantum) physics. 1 Quick overview over the Banach Gelfand Triple S L2 S0 d ∗ ( 0, , 0)(R ); including w -convergence; 2 Convolution and pointwise multiplication; 3 System Theory: impulse response and transfer function; 4 Shannon’s Sampling Theory (>> CD player); p d 5 Fourier transform deﬁned on L (R ); 6 Spectrum and Spectral Analysis Problem;

7 S S0 S0 2d Kernel Theorems; L( 0, 0) ≡ 0(R ); ∗ 8 Soft transitions and w -convergence.

Obviously Fourier Analysis is a very classical ﬁeld, about 200 years old (J.B.Fourier published his seminal paper in 1822). Since the arrival of Lebesgue integration in the early 20th century the setting appears to be quite well understood: Most books in the ﬁeld follow a very similar pattern:

Fourier series >> Fourier transforms >> FFT

Sometimes it is argued, that the FT can be even extended to generalized functions, so-called distributions, e.g. using the Schwartz theory of tempered distributions. Abstract Harmonic Analysis emphasizes the fact that one can do Fourier Analysis over any LCA (locally compact Abelian) group G. which always carries a (unique) Haar measure, and has a dual group Gb. According to Pontryagin Gbb ≡ G. THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Hans G. Feichtinger INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. L1 d The role of (R ), k · k1

Common to most presentations of the Fourier transform (now we restrict our attention to the Euclidean setting) is the (somehow) natural view-point that the FOURIER TRANSFORM is an INTEGRAL transform, given by the following pair of (mutually inverse) formulas: Z −2πis·t d fˆ(s) = f (t) · e dt, t, s ∈ R (5) Rd Here s · t denotes the Euclidean scalar product of the two vectors d s, t ∈ R . The inverse Fourier transform then takes the form: Z f (t) = fˆ(ω) · e2πit·ω dω, (6) Rd

Strictly speaking this inversion formula only makes sense under the 1 d additional hypothesis that fˆ ∈ L (R ), which is not satifsied for 1 d 1 d arbitrary functions f ∈ L (R ). In the general case (f ∈ L (R )) one can obtain f from fˆ using classical summability methods, convergent in the L1-norm. (cf. for example Chap. 1 of [5]). One often speaks of Fourier analysis being the ﬁrst step, telling us how much energy of f is concentrated at a given frequency ω (namely |fˆ(ω)|2, and the Fourier inversion as a method to build f from the pure frequencies (we talk of Fourier synthesis). Unfortunately (unlike the case of Fourier series and periodic functions) the individual terms, namely the pure frequencies, do 1 d 2 d not to the spaces where f lives (L (R ) or L (R )).

One of the (proclaimed) important properties of the Fourier transform is the so-called convolution theorem, i.e. the fact that it turns the somewhat mysterious operation called CONVOLUTION into ordinary pointwise multiplication, i.e. we have

1 d f[∗ g = fˆ · gˆ, f , g ∈ L (R ). (7)

Again it appears most natural to require that two functions to be 1 d convolved should both belong to L (R ), because this ensures that convolution products can be deﬁned pointwise almost everywhere: Z Z f ∗ g(x) := g(x − y)f (y)dy = g(u)f (x − u)du. (8) Rd Rd

Aside from probability (cf. above) convolution has its role in Linear Systems Theory, for the description of time-invariant linear systems, meaning linear operators T , g = T (f ), with time-invariance: T ◦ Tx = Tx ◦ T , ∀x ∈ G. Theorem (hgfei)

Any bounded and translation invariant operator on d C0(R ), k · k∞ (so-called BIBOS system) which commutes with translation is a moving average by some bounded measure, i.e. by d some element in the dual space of C0(R ), k · k∞ . In fact, µ(f ) = Tf (0) describes the system, given by

Tf (x) = [T−x Tf ](0) = T (T−x f )(0) = µ(T−x f = Tx µ(f ).

This pairing establishes an isometric bijection between the dual d space of C0(R ) and the TILS, which is isometric, i.e. with THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Hans G. Feichtinger kµkMb = |kT |kC0 .

Since δz (f ) = f (z) corresponds to the operator Tf = T−z f the usual deﬁnition of such a pairing involves in addition the so-called ﬂip operator f X(x) = f (−x), extended to bounded measures, in order to be compatible with the usual conventions for convolutions INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. Convolution and Fourier Stieltjes transforms

The fact, that the collection of all TILS is not only a Banach space, but also a Banach algebra under composition of operators (with the operator norm) allows to transfer this composition rule to the generating measures. M d One can shown, that ( b(R ), k · kMb ) is then a Banach algebra with respect to convolution imposed in this way, containing 1 d L (R ), k · k1 as a closed, translation-invariant ideal. The collection of characters, i.e. the functions χs are the joint eigenvectors to all these operators, among them the translation operators. The Fourier transform extends to all of these characters, and is then often called Fourier Stieltjes transform, again with

µ\1 ∗ µ2 = µc1 · µc2, d for µ1, µ2 ∈ Mb(R ) (Convolution Theorem for measures).

This approach to convolution can be carried out without the use of measures theory (!), details can be found in my course notes.

In an engineering terminology the measure µ describing the linear system T via T (f ) = µ ∗ f is the impulse response of the system. It can be obtained (!proof) as a w ∗-limit of input functions tending to the Dirac measure, e.g. compressed, normalized (in the L1-sense)rectangular pulses. The Fourier (Stieltjes) transform of the system T is know as the transfer function of the system T , and it is characterized by the eigen-vector property:

µ(χs ) =µ ˆ(s)χs , s ∈ R.

2 d Admittedly the Hilbert space L (R ), k · k2 also plays a very important role for Fourier analysis, thanks to the Plancherel Theorem, telling us (in a sloppy way), that the Fourier transform 2 d can be viewed as a unitary automorphism of L (R ), k · k2 , i.e. a mapping which preserves the L2-norm (and hence scalar products 2 d between elements L (R )).

2 d kfbk2 = kf k2, f ∈ L (R ). (9)

According to engineering terminology the Fourier transform is an energy preserving linear transformation. When we have to prove this theorem on resorts to a veriﬁcation of 1 2 d (9) for f ∈ L ∩ L (R ) and then extends the integral 2 d transform in an abstract way to all of L (R ), k · k2 , which is then shown to be a unitary automorphism. THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Hans G. Feichtinger INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. BANACH GELFAND TRIPLES: a new category

Deﬁnition A triple, consisting of a Banach space B, which is dense in some Hilbert space H, which in turn is contained in B0 (hence w ∗-dense there) is called a Banach Gelfand triple.

Deﬁnition B B0 B B0 If ( 1, H1, 1) and ( 2, H2, 2) are Gelfand triples then a linear operator T is called a [unitary] Gelfand triple isomorphism if

1 A is an isomorphism between B1 and B2.

2 A is a unitary isomorphism between H1 and H2. ∗ 3 A extends to a weak isomorphism as well as a norm-to-norm B0 B0 continuous isomorphism between 1 and 2.

In principle every CONB (= complete orthonormal basis) Ψ = (ψi )i∈I for a given Hilbert space H can be used to establish such a unitary isomorphism, by choosing as B the space of elements within H which have an absolutely convergent expansion, P i.e. satisfy i∈I |hx, ψi i| < ∞. For the case of the Fourier system as CONB for H = L2([0, 1]), i.e. the corresponding deﬁnition is already around since the times of N. Wiener: A(T), the space of absolutely continuous Fourier series. It is also not surprising in retrospect to see that the dual space 0 PM(T) = A(T) is space of pseudo-measures. One can extend the classical Fourier transform to this space, and in fact interpret this extended mapping, in conjunction with the classical Plancherel theorem as the ﬁrst unitary Banach Gelfand triple isomorphism, 2 1 2 ∞ between (A, L , PM)(T) and (` , ` , ` )(Z).

A couple of basic facts resulting direct from the definition, combined with the general fact that the space of w ∗-continuous linear functionals on a dual space B0 coincides naturally with the original Banach space, gives the following facts. S L2 S0 We apply them to the concrete Banach Gelfand Triple ( 0, , 0). For every BGTr-mapping there is an adjoint BGTr-mapping T ∗, and T ∗∗ = T for each of them; 2 d d If a bounded linear mapping on L (R ), k · k2 leaves S0(R ) invariant and so does T ∗ (the Hilbert adjoint), then T (and also T ∗) define BGTr-homomorphism. Among others this principle can be used to show that the even fractional Fourier transforms define (unitary) BGTr- S L2 S0 d automorphisms on ( 0, , 0)(R ).

There are situations, where one is forced to view even more general situations than BGTr-operators (BGOs), even if one is a priori interested only in the Hilbert space behaviour: Here are some examples: 2 d d d 1 Let g ∈ L (R ) be given, and any lattice Λ C R × Rb . Then the coeﬃcient operator

C : f 7→ (Vg f (λ))λ∈Λ

2 d d 2 only maps L (R ) into cosp(Λ), but S0(R ) into ` (Λ). 2 similar situation for the synthesis operator, which is its adjoint. 3 Finally the frame-operator is a priori an operator which maps d 0 d S ( ), k · kS into (S ( ), k · kS0 ). 0 R 0 0 R 0 But this is good enough to show that it has a Janssen representation. THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Hans G. Feichtinger INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. Regularizing Operators

THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory HansAbbildung: G. Feichtinger BGTREG1.jpg INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. Regularizers help to approximate distributions

An important family of regularizing operators are those bounded 0 d families of BGOs, where each of them maps (S ( ), k · kS0 ) into 0 R 0 S d 0(R ), k · kS0 , and which form an approximating the identity. Abstractly speaking they form bounded nets of BGOs which S d converge strongly (e.g. in their action on 0(R ), k · kS0 ) to the identity. Since we have

S0 S S S S0 S S S ( 0 ∗ 0) · 0 ⊂ 0 and ( 0 · 0) ∗ 0 ⊂ 0

this can be product-convolution (or convolution-product) operators. A similar property is shared by Gabor multipliers with ﬁnitely many symbols (using lattice points in a bounded domain).

d The Fourier transform F on R has the following properties: d d 1 F is an isomorphism from S0(R ) to S0(Rb ), 2 d 2 d 2 F is a unitary map between L (R ) and L (Rb ), 3 F is a weak* (and norm-to-norm) continuous bijection from S0 d S0 d 0(R ) onto 0(Rb ). Furthermore, we have that Parseval’s formula

hf , gi = hfb, gbi (10) S d S0 d is valid for (f , g) ∈ 0(R ) × 0(R ), and therefore on each level S L2 S0 d of the Gelfand triple ( 0, , 0)(R ).

S L2 S0 d It is not diﬃcult to show, that the norms of ( 0, , 0)(R ) 1 2 ∞ 2d correspond to norm convergence in (L , L , L )(R ). The FOURIER transform, viewed as a BGT-automorphism is uniquely determined by the fact that it maps pure frequencies χω(t) = exp(2πiωt) onto the corresponding point measures δω. Vice versa, (even non-harmonic) trigonometric polynomials are S0 d those elements of 0(R ) which have ﬁnite support on the Fourier transform side.

1 d To know that an f ∈ L (R ) has a continuous Fourier transform d fb ∈ C0(R ) is nice, and also to know that the Fourier transform is linear and injective, with a dense, but proper range named L1 d C d F (R ), k · kFL1 ,→ 0(R ), k · k∞ (Lemma of Riemann-Lebesgue), but how can one recover the function from its 1 d Fourier transform for general f ∈ L (R ) (knowing that it may 1 d happen that fb ∈/ L (R )!!) In this situation a simple abstract result helps: Lemma d Given h ∈ S0(R ) with h(0) = 1, then one has: d fρ : s 7→ h(ρs) · fb(s) is in S0(R ) for any ρ > 0, and therefore the inverse Fourier transform is well deﬁned in the pointwise sense (even using Riemann integration). −1 1 d Furthermore F fρ → f in L (R ), k · k1 , for ρ → 0.

Arguments supporting the above lemma: d First of all we ﬁnd that the dilation invariance of S0(R ) d implies that s 7→ h(ρs) belongs to S0(R ) as well; 1 d d d Since L (R ) ∗ S0(R ) ⊆ S0(R ) (with corresponding norm estimates), one has on the Fourier transform side

1 d d d FL (R ) · S0(R ) ⊂ S0(R ).

−1 R Finally g = F (h) satisﬁes d g(t)dt = 1, and hence one R d has a bounded approximate unit (within S0(R )) convolved with f (on the time-side), hence norm convergence in 1 d L (R ), k · k1 . The same reasoning even implies that one has convergence in p d L (R ), k · kp for 1 ≤ p ≤ 2, because one ﬁnds that even p d d q d d 1 d FL (R ) · S0(R ) ⊂ L (R ) · S0(R ) ⊂ L (R ). THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Hans G. Feichtinger INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. S d Classical Summability Kernels ARE in 0(R ) The above lemma is of course only useful if one can check that S d concrete kernels belong to 0(R ), k · kS0 . For the one-dimensional case this has been investigated by F. Weisz.

THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Hans G.Abbildung: Feichtinger weisz8.jpg INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. Why should we care about convolutions?

In engineering, especially in communication theory, the theory of time-invariant channels, i.e. operators which are linear and commute with translations (motivated by the time-invariance of physical laws!), also called TILS (time-invariant linear systems) is one of the cornerstones. The collection of all these operators form a commutative algebra, with the pure frequencies as common eigenvalues. Still in an engineering terminology: Every such system T is a moving average or (equivalently) a convolution operator, described by the impulse response of the system, or alternatively, it can be described as a Fourier multiplier, being understood as a pointwise multiplier on the Fourier transform side, by the so-called transfer function.

From my point of view the most natural setting is that of so-called BIBOS, bounded-input-bounded-output systems, or more precisely d the bounded linear operators on C0(R ), k · k∞ which commute with translations. It is not hard to show that they are exactly the convolution operators which are arising from a given linear functional d 0 d µ ∈ M ( ) := C ( ), k · kC 0 . b R 0 R 0 d Given µ ∈ Mb(R ) we can deﬁne a convolution operator via

X d µ ∗ f (x) = µ(Tx f ), x ∈ R , (11)

d where f X(x) = f (−x), x ∈ R . d Any TILS on C0(R ) is if this form: Given T one ﬁnds µ d by the rule µ(f ) := [T (f X)](0), f ∈ C0(R ).

According to R. Larsen (his book on the multiplier problem appeared in 1972) it is meaningful to ask for a characterization of the space of “multipliers”, i.e. all linear, bounded operators from one translation invariant Banach space into another one, which commute with translations. You may think of such operators from p d q d L (R ), k · kp to L (R ), k · kq , for two values p, q ∈ [1, ∞]. Using the concept of quasi-measures introduced by G. Gaudry he demonstrates that each such operator is a convolution operator by some quasi-measure, which also has an alternative description as a Fourier multiplier. The drawback of the concept of quasi-measures is the fact, that they do not involve any global condition and thus a general quasi-measure does NOT have a well deﬁned Fourier transform. So the expected claim that the transfer function is the FT of the impulse response (and vice versa) cannot be formulated in that context. THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Hans G. Feichtinger INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. Multipliers between Lp-spaces

d The key result for multipliers in the context of S0(R ) can be summarized as follows: Theorem S d Any bounded linear operator from 0(R ), k · kS0 into 0 d (S ( ), k · kS0 ) which commutes with translations can be 0 R 0 S0 d described by the convolution with a uniquely σ ∈ 0(R ), i.e. via

X d d Tf (x) = σ(Tx [f ]), f ∈ S0(R ), x ∈ R .

d d Thus in fact, T maps S0(R ) even into Cb(R ), k · k∞ , with norm equivalence between the operator norm of the operator T 0 d and the functional norm (in (S ( ), k · kS0 )) of σ. 0 R 0 If σ is a regular distribution, induced by some test function d h ∈ S0(R ), then we even have that f 7→ h ∗ f (equivalently given 0 d d in a pointwise sense) maps (S ( ), k · kS0 ) into C ( ), k · k . 0 R 0 b R ∞ THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Hans G. Feichtinger INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. Why Linear Functional Analysis

1 When dealing with continuous variables most of the function spaces V that we can think of (even the vector space of all polynomials over R!) are NOT FINITE dimensional; 2 Hence we have to talk about limits of finite linear combinations, i.e. convergence in some norm or at least topology, i.e. we need (w.l.g.) Banach spaces and topological vector spaces; 3 Since we cannot describe everything with any fixed basis we have to work with the family of all possible coordinate systems for all possible finite-dimensional subspaces, so in fact with the dual space (constituted by all bounded linear functionals on the given space.

1 Theory of Hilbert spaces, orthonormal bases; 2 The Hahn-Banach Theorem, reﬂexivity, ∗ 3 w -convergence; the Banach-Alaoglou Principles 4 The Closed-Graph Theorem, Open Mapping Theorem; 5 Banach Steinhaus Theorem

2 1 The Hilbert space L (T), k · k2 , with Fourier series; p d 2 The spaces L (R ), k · kp , wich are reﬂexive for 1 < p < ∞; 3 Existence of solutions in the weak sense (PDE); 4 Automatic continuous embeddings, uniqueness of norms; 5 Convergence of sequences of functionals; 6 Boundedness of maximal operator and pointwise a.e. convergence;

1 Lebesgue’s theory of the integral is still at the basis of Fourier analysis when viewed as an integral transform, with e.g. questions about pointwise convergence and Lp-spaces; 2 Calderon-Zygmund theory of singular integrals brought up the importance of BMO and the real Hardy spaces, with their atomic characterization and maximal functions; 3 interpolation theory, e.g. for Hausdorﬀ-Young; 4 Study of diﬀerentiability, singular integrals etc. let to the theory of function spaces (E. Stein, J. Peetre, H. Triebel).

One of the big new tools in the second half of the 20th century was certainly the theory of tempered distributions introduced by Laurent Schwartz in the 50th. 1 It gives a (generalized) Fourier transform to many objects which did not “have a FT” so far; 2 It provides a basis for the solution of PDEs (H¨ormander); 3 It makes use of nuclear Frechet spaces, e.g. in order to prove the kernel theorem; It thus also had a strong inﬂuence of the systematic development of the theory of topological vector spaces spaces (J. Horvath (1966), F. Treves (1967), H. Schaefer (1971)).

In the context of tempered distributions many things became “kind of simple”. Instead of pointwise convergence on can talk about distributional convergence, most operations (including the Fourier transform) are indeed continuous, and since the convergence is d relatively weak the space S(R ) of test functions is dense in 0 d S (R ), hence all the operations can be viewed as the natural extension of the classical concepts. More pratically speaking, using duality theory one can deﬁne

d σb(f ) = σ(fb), f ∈ S(R ), d but alternatively one could take any sequence hn in S(R ) convergent (distributionally) to σ and deﬁne

σ = lim hbn. b n→∞

S0 d One can regularize distributions from 0(R ) using Wiener amalgam convolution and pointwise multiplier results:

S S0 S S S S0 S S 0 · ( 0 ∗ 0) ⊆ 0, 0 ∗ ( 0 · 0) ⊆ 0 (12) S0 This means, convolution of a mild distribution σ ∈ 0 with some test function gives a function which is locally in the Fourier algebra and has enough “smoothness” so that it becomes a pointwise S d multiplier of 0(R ), k · kS0 . S0 S d On the other hand, multiplying σ ∈ 0 by some h ∈ 0(R ) provides enough decay (in the sense of global summability in an 1 d ` -sense) so that subsequent convolution by f ∈ S0(R ) produces again test functions.

In his studies Norbert Wiener considered the Banach algebra A(T), k · kA of absolutely convergent Fourier series. It was one of the early Banach algebras, with Wiener’s inversion Theorem being an important first example. Later on it was natural to study the dual space, which of course contains the dual space of C(T), k · k∞ , which by the Riesz representation theorem can be identified with the bounded (regular Borel) measures on the torus it was natural to call these functions pseudo-measures. 1 Since A(T) can be identified with L (T) (viewed as subspaces of 2 2 L (T) and ` (Z) respectively), it is natural to expect (and prove ∞ distributionally) that PM(T) is isomorphic to ` (Z) via the (extended) Fourier transform.

The case a · b < 1 has further good properties: 2 2 1 The minimal norm solution in ` (Z ) depends linearly and 2 continuously on f in L (R), k · k2 . 2 For f ∈ S(R) the Gabor series is even convergent in the topology of the Schwartz space (with rapidely decreasing coeﬃcients); 3 It is also possible to replace the Gauss function by another Gauss-like function h (for each ﬁxed a, b with ab < 1) such that one has the following2 quasi-orthogonal expansion:

X 2 f = hf , hk,nihk,n, ∈ L (R) (13) k,n

2 with unconditional convergence in L (R), k · k2 . 2!ad hoc terminology! THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Hans G. Feichtinger INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. Natural properties

This observation also implies that S d Lp d 0(R ), k · kS0 ,→ (R ), k · kp for any p ≥ 1, or any other d function/distribution space on R with isometric TF-shifts. S d The minimality of 0(R ), k · kS0 gives, on the other hand, the Lq d S0 d continuous embedding of any (R ), k · kq into 0(R ). An emerging theory of Fourier standard spaces is on the way. They satisfy S d B S0 d 0(R ) ,→ ,→ 0(R ), and some further conditions, such as

1 d 1 d L (R ) ∗ B ⊂ B and FL (R ) · B ⊂ B.

The minimality implies (inspired by the atomic characterization of real Hardy spaces the following result: Theorem

d Given any non-zero g ∈ S0(R ) one has

d X X S0(R ) = ck π(λk )g, with |ck | < ∞ . k≥1 k≥1

This results also implies that the space is invariant under dilation, rotation and even fractional Fourier transforms. A long list of suﬃcient conditions ensures that among others all d classical summability kernels belong to S0(R ).

H. G. Feichtinger and G. Zimmermann. A Banach space of test functions for Gabor analysis. In H. G. Feichtinger and T. Strohmer, editors, Gabor Analysis and Algorithms: Theory and Applications, Applied and Numerical Harmonic Analysis, pages 123–170. Birkh¨auser Boston, 1998. J. J. Benedetto, C. Heil, and D. F. Walnut. Gabor systems and the Balian-Low theorem. In Gabor Analysis and Algorithms: Theory and Applications, Appl. Numer. Harmon. Anal., pages 85–122. Birkh¨auser Boston, Boston, MA, 1998. H. G. Feichtinger and N. Kaiblinger. Varying the time-frequency lattice of Gabor frames. Trans. Amer. Math. Soc., 356(5):2001–2023, 2004. H. G. Feichtinger and F. Weisz. Inversion formulas for the short-time Fourier transform. J. Geom. Anal., 16(3):507–521, 2006.

2 d Combined with the Hilbert space L (R ), k · k2 (in the middle) S d L2 d S0 d the three spaces 0(R ), (R ) and 0(R ) form a so-called Banach Gelfand Triple or rigged Hilbert space. S d Such a BGTr is constituted by a Banach space ( 0(R ), k · kS0 in our case) which is continuously and densely embedded into a 2 d Hilbert spaces H = L (R ) in our case) and such that this itself is embedded continuously into the dual Banach space (norm continuous, but only w ∗-dense). A morphism of Banach Gelfand triples respects each of the three layers, but is also w ∗-w ∗--continous at the outer level.

The Fourier transform is a prototype of a unitary BGTr-automorphism for the Banach Gelfand Triple S L2 S0 d ( 0, , 0)(R ). It can be formulated for any LCA group, and reduces in the classical case of G = T to a well known result, making use of A(T), k · kA , the algebra of absolutely convergent Fourier series and its dual, the space PM of pseudo-measures. The classical Fourier transform, establishing a unitary isomorphism 2 2 (Parseval’s identity) from L (T), k · k2 to ` (Z) extends to a 2 BGTr isomorphism between (A(T), L (T), PM(T)) and 1 2 ∞ (` , ` , ` )(Z).

p d 0 d The fact that L ( ), k · k ,→ (S ( ), k · kS0 ) for any R p 0 R 0 1 ≤ p ≤ ∞ implies that one go beyond Hausdorff-Young, which p d q d shows that for 1 ≤ p ≤ 2 on has FL (R ) ⊂ L (R ) (with 1/p + 1/q = 1). Even for p > 2 there is a Fourier transform (locally in PM = FL∞). p d 0 d The (known) statement that FL (R ) ⊂ S (R ) is of course much Lp d S0 d weaker than the claim F (R ) ⊂ 0(R ). ∞ d This fact allows to directly define the spectrum of h ∈ L (R ) using the general definiton:

S0 spec(σ) := supp(σb), σ ∈ 0. If supp(σ) is a ﬁnite set it can be shown to be just a ﬁnite sum PK of Dirac measures, i.e. σ = k=1 ck δxk .

S0 d Putting ourselves in the setting of 0(R ) (which contains all of ∞ d d L (R ), hence Cb(R ) and even more special the pure frequencies, i.e. the exponential functions χs (t) = exp(2πis · t), for d t, s ∈ R ), we may ask ourselves the following two questions:

1 What are the pure frequencies which can be “ﬁltered out of C d S0 d the signal”( h ∈ b(R ) or σ ∈ 0(R )); 2 Can one resynthesize the signal from those pure frequencies (as it is obvious of the case of classical Fourier series expansions of periodic functions);

It is clear from the prototypical BGTr (`1, `2, `∞) that the Hilbert space (here `2) is not always dense in the dual of the Banach space, here `∞. However, every element in `∞ can be approximated in the coordinate-wise sense by a bounded sequency in the closed subspace cosp, which is the closure of `2 within ∞ ` , k · k∞ . Note that the situation is diﬀerent e.g. in the context of Sobolev space, where for s > 0 one has

d 2 d d Hs (R ) ,→ L (R ) ,→ H−s (R )

with dense embeddings. We will be mostly concerned with sequences which are ∗ d w -convergent. Over R this simpliﬁcation is fully justiﬁed, S d because 0(R ), k · kS0 is a separable Banach space.

So let us ﬁrst recall what on the one hand norm-convergence and ∗ S0 d w -convergence within 0(R ) in concrete terms, thanks to to the S d atomic characterization of 0(R ), k · kS0 : Lemma

d Let 0 6= g ∈ S0(R ) be given. Then one has: 0 d S0 d 0 d Given σ ∈ S (R ), then belongs to 0(R ) ,→ S (R ) if and only if the (continuous) function Vg (σ) = σ(gλ) is bounded, and: S0 d sup d d |Vg (σ)(λ)| deﬁnes an equivalent norm on ( ). λ∈R ×Rb 0 R S0 d ∗ σ0 ∈ 0(R ) is the w -limit of a sequence (σn)n≥1 if and only if one has (uniformly over compacts):

lim Vg (σn)(λ) = Vg (σ0)(λ). n→∞

In order to demonstrate the possibility of approximating a given S0 ∗ distribution σ ∈ 0 in the w -sense by discrete measures we have to look at (quasi)-interpolation operators acting on S d 0(R ), k · kS0 . It is clear that by compressing the sampling grid one has uniform convergence for f ∈ C0(R), but in fact on can show the non-trivial S S fact that one has convergence in 0(R), k · kS0 for f ∈ 0(R) S (note that ∆ ∈ 0(R) and the sampling sequence (f (αn))n∈Z is in 1 ` (Z). Hence the PLI, which is an absolutely sum of the form h = P f (αn)T ∆ belongs to S ( ), k · kS and the n∈Z αn α 0 R 0 question of convergence is at least meaningful. Similar arguments hold for e.g. cubic B-splines (which also are in S0(R) and form a BUPU Ψ = (ψn)n∈Z. By taking tensor products one can of course ﬁnd multi-variate P BUPUs, with small support and d ψ (x) ≡ 1. n∈Z n THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Hans G. Feichtinger INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. w ∗-density of discrete measuresII

S0 Consequently, given σ ∈ 0, the action of the adjoint operators converge in the w ∗-sense to σ. We just have to verify that the adjoint action can be desribed by the operator

α X α DΨ(σ) = σ(ψn )δαn. n∈Zd

0 d This family is also uniformly bounded in (S ( ), k · kS0 ) (for 0 R 0 α → 0).

piecewise linear interpolation as a sum of triangular functions

0.2

0.15

0.1

0.05

50 100 150 200 250 300 350 400 450

smooth functions and sampling values

0.2

0.15

0.1

0.05

50 100 150 200 250 300 350 400 450 THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Abbildung: PiecewiseHans G. linear Feichtinger interpolation using triangular BUPU

For bounded measures this operator (with the simple partition using box-car functions) can be interpreted as the approximation of a probability measure through the corresponding histograms. INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. A Cauchy approach to distributions

The simple setting of Banach spaces (resp. dual spaces) allows also to work with a very simple, alternative way to introduce the dual space, namely to deﬁne

1 weak Cauchy-sequences( hn)n≥1 of test functions from d S0(R ), in the sense that d (hh, fni)n≥1 is a Cauchy sequence in C for any h ∈ S0(R ); 2 then define equivalence classes of such sequences; 3 then define the norm of such a class (it is finite by Banach-Steinhaus). It is then not difficult to show that this is just another way (maybe 0 d more intuitive to engineers) to describe (S ( ), k · kS0 )! 0 R 0

So in the terminology of “mild distributions” one can prove and has the follwing statement: Lemma

An element s ∈ Rcd in the frequency domain belongs to the ∞ d spectrum of h ∈ L (R ) resp. to supp(hb) if and only if the ∗ corresponding pure frequency χs can be obtained as the w -limit of a sequence of convolution products hn := h ∗ fn, for fn being d chosen suitably in S0(R ).

1 ∞ d Whenever h ∈ L ∩ L (R ) we know that hb is a continuous function. Then it is easy to ﬁlter χs out of h if hb(s) 6= 0 (just take modulated Fejer kernels). When s belongs to the closure of this set one has to be a bit more careful!

Now of course the question of spectral synthesis, i.e. the ∞ d reconstruction of σ (or even h ∈ L (R )) from the pure frequencies which have been “found within” h appears to be like a natural task. If we ask the question for the space of bounded measures d 0 d (M ( ), k · kM ) which we define as C ( ), k · kC 0 then there b R b 0 R 0 is also a well-defined support, and one can show (using properties d of C0(R ), k · k∞ !) that any bounded measure can be described as the w ∗-limit of a sequence of finite (discrete) measures of the K n n form µ = P c δ n , with x ∈ supp(µ). n k=1 k xk k S0 d That a similar property is NOT valid for general σ ∈ 0(R ), and ∞ d in fact not even for h ∈ L (R ), for d ≥ 3, is one of the remarkable facts of Fourier Analysis.

I cannot go into the details of the description of the spectrum of ∞ d h ∈ L (R ) as given in the book of Reiter, but just want to recall some building blocks there. ∞ d First, h ∈ L (R ) is considered as a convolution operator from 1 d ∞ d d L (R ) into L (R ) (in fact Cb(R )). This convolution operator has a well deﬁned kernel (null-space), which in fact is a 1 d (translation-invariant) and closed ideal I = I (h) in L (R ), k · k1 . The set of all points in the Fourier domain where some f ∈ I has non-vanishing FT is called the cospectrum of I . The complement of this (open) set is then deﬁned as the spectrum of h. It is one of the interesting

In the Euclidean Situation interesting objects are for example the d Dirac trains supported by lattices Λ C R : X tt := δλ. λ∈Λ

S0 d These (unbounded, discrete) measures belong to 0(R ) and thus have a distributional Fourier transform. In fact, the usual argument in distribution theory making use of Poisson’s formula (now not d d d only for f ∈ S(R ) but for any f ∈ S0(R )!) gives (Λ = Z ):

ttc = tt. (14) This fact allows to prove Shannon’s sampling Theorem (based on intuitive arguments).

This result is the key to prove Shannon’s Sampling Theorem which is usually considered as the fundamental fact of digital signal processing (Claude Shannon: 1916 - 2001). Shannon’s Theorem says that one can have perfect reconstruction for band-limited functions. If the so-called Nyquist criterion is satisﬁed (sampling distance small enough), i.e. supp(fˆ) ⊂ [−1/α, 1/α], then

X d f (t) = f (αk)g(x − αk), x ∈ R . (15) k∈Zd For more general sampling lattices the support size conditions have to avoid overlapping of the periodization of the spectrum with respect to the orthogonal lattice Λ⊥.

a lowpass signal, of length 720 its spectrum, max. frequency 23 0.2

0.05 0.1

0 0 −0.1 −0.05 −0.2 −200 0 200 −200 0 200

the sampled signal , a = 10 the FT of the sampled signal

0.03 0.05 0.02 0.01 0 0 −0.01 −0.05 −0.02 −200 0 200 −200 0 200

The KERNEL THEOREM for S(Rd )

The kernel theorem for the Schwartz space can be read as follows: Theorem d 0 d For every continuous linear mapping T from S(R ) into S (R ) 0 2d there exists a unique tempered distribution σ ∈ S (R ) such that

d T (f )(g) = σ(f ⊗ g), f , g ∈ S(R ). (16)

0 2d Conversely, any such σ ∈ S (R ) induces a (unique) operator T such that (16) holds.

d The proof of this theorem is based on the fact that S(R ) is a nuclear Frechet space, i.e. has the topology generated by a sequence of semi-norms, can be described by a metric which d turns S(R ) into a complete metric space.

The Schwartz Kernel Theorem extends a characterization of L2-kernel-operators of the form to the most general level. It shows 2 2d that for any “integral kernel” K(x, y) ∈ L (R ) the following integral operator (continuous analogue of matrix multiplication) Z d TK f (x) = K(x, y)f (y)dy, x ∈ R . (17) Rd 2 d deﬁnes a bounded, in fact compact operator on L (R ), with singular values in `2, i.e. a Hilbert-Schmidt operators. Altogehter they form a Hilbert space with respect to the scalar product

∗ hT , SiHS := trace(TS ), (18)

p ∗ and corresponding norm kT kHS := trace(TT ). With this scalar product we have a unitary isomorphism 2 2d between (HS, k · kHS ) and L (R ), k · k2 . THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Hans G. Feichtinger INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. S L2 S0 2d Extending this situation to ( 0, , 0)(R )

If this unitary isomorphism can be extended to a Banach Gelfand Triple isomorphism we have to look what on can say about kernels S L2 S0 2d in ( 0, , 0)(R ). Of course we look ﬁrst what one can say for good kernels 2d K(x, y) ∈ S0(R ), which (hopefully, and in fact) behave like “continuous matrices”. Given such a kernel it is w ∗-to-norm S0 d S d continuous from 0(R ) to 0(R ), k · kS0 and it is possible to d deﬁne T (δy ) ∈ S0(R ). In fact, this one gets (as expected from matrix multiplication)

T (δy )(x) = K(x, y)

in analogy to the case TA(x) = A ∗ x, with

aj,k = hTA(ek ), ej i.

The KERNEL THEOREM for S0 I

Tensor products are also most suitable in order to describe the set of all operators with certain mapping properties. The backbone of the corresponding theorems are the kernel-theorem which reads as S d follows (!! although 0(R ), k · kS0 is NOT a nuclear space) One of the corner stones for the kernel theorem is: the tensor-product factorization: Lemma

k n ∼ k+n S0(R )⊗ˆ S0(R ) = S0(R ), (19) with equivalence of the corresponding norms.

d For G = R it follows readily from the characterization using the atomic decomposition using Gaussians.

The KERNEL THEOREM for S0 II S S0 The Kernel Theorem for general operators in L( 0, 0): Theorem S d S0 d If K is a bounded operator from 0(R ) to 0(R ), then there S0 2d exists a unique kernel k ∈ 0(R ) such that hKf , gi = hk, g ⊗ f i d for f , g ∈ S0(R ), where g ⊗ f (x, y) = g(x)f (y). Formally sometimes one writes by “abuse of language” Z Kf (x) = k(x, y)f (y)dy Rd with the understanding that one can deﬁne the action of the S0 d functional Kf ∈ 0(R ) as Z Z Z Z Kf (g) = k(x, y)f (y)dy g(x)dx = k(x, y)g(x)f (y)dxdy. Rd Rd Rd Rd THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Hans G. Feichtinger INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm.

The KERNEL THEOREM for S0 III

This result is the “outer shell” of the Gelfand triple isomorphism. The “middle = Hilbert” shell which corresponds to the well-known 2 d result that Hilbert Schmidt operators on L (R ) are just those compact operators which arise as integral operators with 2 2d L (R )-kernels. This description can be also extended to the Kohn-Nirenberg symbol of an operator (no functions or distributions over d d R × Rb ), or alternatively the Weyl-symbol (in the sense of pseudo-diﬀerential operators), and ﬁnally the spreading representation. The symplectic Fourier transform (a further unitary BGTr isomorphism) the transfers the information between the spreading function η(T ) and the Kohn-Nirenberg symbol σ(T ).

The KERNEL THEOREM for S0 IV

Theorem The classical kernel theorem for Hilbert Schmidt operators is 0 unitary at the Hilbert spaces level, with hT , SiHS = trace(T ∗ S ) as scalar product on HS and the usual Hilbert space structure on 2 2d L (R ) on the kernels. 2d Moreover, such an operator has a kernel in S0(R ) if and only if S0 d S d the corresponding operator K maps 0(R ) into 0(R ), but not only in a bounded way, but also continuously from w ∗−topology d into the norm topology of S0(R ). In analogy to the matrix case, where the entries of the matrix

ak,,j = T (ej )k = hT (ej ), ek i

we have for K ∈ S0 the continuous version of this principle: d K(x, y) = δx (T (δy ), x, y ∈ R . THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Hans G. Feichtinger INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. The Kernel Theorem as a BGT isomorphism

The diﬀerent version of the kernel theorem for operators between S S0 0 and 0 can be summarized using the terminology of Banach Gelfand Triples (BGTR) as follows. Theorem There is a unique Banach Gelfand Triple isomorphism between the S L2 S0 2d Banach Gelfand triple of kernels ( 0, , 0)(R ) and the operator Gelfand triple around the Hilbert space HS of Hilbert Schmidt S0 S S S0 operators, namely (L( 0, 0), HS, L( 0, 0)), where the ﬁrst set is ∗ S0 d understood as the w to norm continuous operators from 0(R ) d to S0(R ), the so-called regularizing operators.

Let us know motivate how the BGTr setting can be used to provide a good intutive understanding of the spreading representation of general operators. As we will see the spreading representation gives us a complete characterization of operators with kernels S L2 S0 2d K(x, y) ∈ ( 0, , 0)(R ) with the corresponding spreading S L2 S0 d d symbols in η ∈ ( 0, , 0)(R × Rb ). What are the most simple (non-compact, not HS) operators: the pure TF-shift operators π(λ) = Ms Tt should correspond to a Dirac d d at λ = (t, s) ∈ R × Rb . On the other hand one can imagine (and verify) that an operator which can be written as Z H(λ)π(λ)dλ, Rd ×Rbd d d 2d with H ∈ S0(R × Rb ) is a operator with kernel in S0(R ). THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Hans G. Feichtinger INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. Spreading function and Kohn-Nirenberg symbol

The following summary has been provided by Götz Pfander (now Univ. Eichstätt): 1 S0 d For σ ∈ 0(R ) the pseudodifferential operator with Kohn-Nirenberg symbol σ is given by: Z 2πix·ω Tσf (x) = σ(x, ω)fˆ(ω)e dω Rd The formula for the integral kernel K(x, y) is obtained Z Z −2πi(y−x)·ω Tσf (x) = σ(x, ω)e dω f (y)dy Rd Rd Z = k(x, y)f (y)dy. Rd 2 The spreading representation of Tσ arises from ZZ Tσf (x) = σb(η, u)MηT−uf (x)dudη. R2d THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory σb is called theHans spreading G. Feichtinger function of Tσ. INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. Further details concerning Kohn-Nirenberg symbol

(courtesy of Goetz Pfander (Eichst¨att):) y y · Symmetric coordinate transform: Ts F (x, y) = F (x + 2 , x − 2 ) · Anti-symmetric coordinate transform: TaF (x, y) = F (x, y − x)

· Reﬂection: I2F (x, y) = F (x, −y)

· partial Fourier transform in the ﬁrst variable: F1

· partial Fourier transform in the second variable: F2 The kernel K(x, y) can be described as follows:

−1 K(x, y) = F2σ(η, y − x) = F1 σb(x, y − x) Z 2πiη·x = σb(η, y − x) · e dη. Rd

operator H Hf (x) l = R kernel κH κH (x, s)f (s) ds l = R 2πix·ω Kohn–Nirenberg symbol σH σH (x, ω)fb(ω)e dω l = R time–varying impulse response hH hH (t, x)f (x − t) dt l = RR 2πix·ν spreading function ηH ηH (t, ν)f (x − t)e dt dν = RR ηH (t, ν)MνTt f (x), dt dν ,

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THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Hans G. Feichtinger