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FOURIER STANDARD SPACES and the Kernel Theorem

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FOURIER STANDARD SPACES and the Kernel Theorem Numerical Harmonic Analysis Group FOURIER STANDARD SPACES and the Kernel Theorem Hans G. Feichtinger [email protected] www.nuhag.eu . Currently Guest Prof. at TUM (with H. Boche) Garching, TUM, July 13th, 2017 Hans G. Feichtinger [email protected] www.nuhag.euFOURIER. Currently STANDARD GuestSPACES Prof. at TUM and the (with Kernel H. Boche) Theorem OVERVIEW d We will concentrate on the setting of the LCA group G = R , although all the results are valid in the setting of general locally compact Abelian groups as promoted by A. Weil. |||||||||||||||||||||- Classical Fourier Analysis pays a lot of attention to p d L (R ); k · kp because these spaces (specifically for p 2 f1; 2; 1g) are important to set up the Fourier transform as an integral transform which also respects convolution (we have the convolution theorem) and preserving the energy (meaning that it is 2 d a unitary transform of the Hilbert space L (R ); k · k2 ). |||||||||||||||||||||- d Occasionally the Schwartz space S(R ) is used and its dual 0 d S (R ), the space of tempered distributions (e.g. for PDE and d the kernel theorem, identifying operators from S(R ) to 0 d 0 2d S (R ) with their distributional kernels in S (R )). Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem OVERVIEW II S d In the last 2-3 decades the Segal algebra 0(R ); k · kS0 1 d (equal to the modulation space (M (R ); k · kM1 )) and its dual, 0 d 1 d S 0 M ( 0 (R ); k · kS0 ) or (R ) have gained importance for many questions of Gabor analysis or time-frequency analysis. Fourier standard spaces is a new name for a class of Banach d 0 d spaces sandwiched in between S0(R ) and S0 (R ), with two module structures, one with respect to the Banach convolution 1 d algebra L (R ); k · k1 , and the other by pointwise multiplication L1 d with elements of the Fourier algebra F (R ); k · kFL1 . As we shall point out there is a huge variety of such spaces, and many questions of Fourier analysis find an appropriate description in this context. Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem OVERVIEW III The spaces in this family are useful for a discussion of questions in Gabor Analysis, which is an important branch of time-frequency analysis, but also for problems of classical Fourier Analysis, such as the discussion of Fourier multipliers, Fourier inversion questions (requiring to work 1 d 1 d with the space L (R ) \FL (R )), and many other spaces. Within the family there are two subfamilies, namely the Wiener amalgam spaces and the so-called modulation spaces, among them S d the Segal algebra 0(R ); k · kS0 or Wiener's algebra 1 d W (C0; ` )(R ); k · kW . Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem The key-players for time-frequency analysis Time-shifts and Frequency shifts Tx f (t) = f (t − x) d and x; !; t 2 R 2πi!·t M!f (t) = e f (t) : Behavior under Fourier transform ^ ^ (Tx f )b= M−x f (M!f )b= T!f The Short-Time Fourier Transform Vg f (λ) = hf ; M!Tt gi = hf ;π (λ)gi = hf ; gλi; λ = (t;!); Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem A Typical Musical STFT Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Demonstration using GEOGEBRA (very easy to use!!) Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Spectrogramm versus Gabor Analysis Assuming that we use as a \window" a Schwartz function d 2 g 2 S(R ), or even the Gauss function g0(t) = exp(−πjtj ), we can define the spectrogram for general tempered distributions 0 d f 2 S (R )! It is a continuous function over phase space. In fact, for the case of the Gauss function it is analytic and in fact a member of the Fock space, of interest within complex analysis. Both from a pratical point of view and in view of this good smoothness one may expect that it is enough to sample this spectrogram, denoted by Vg (f ) and still be able to reconstruct f (in analogy to the reconstruction of a band-limited signal from regular samples, according to Shannon's theorem). Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem So let us start from the continuous spectrogram 2 d The spectrogram Vg (f ), with g; f 2 L (R ) is well defined and has a number of good properties. Cauchy-Schwarz implies: 2 d kVg (f )k1 ≤ kf k2kgk2; f ; g 2 L (R ); d d in fact Vg (f ) 2 C0(R × Rb ). Plancherel's Theorem gives 2 d kVg (f )k2 = kgk2kf k2; g; f 2 L (R ): 2 d Assuming that g is normalized in L (R ), or kgk2 = 1 makes f 7! Vg (f ) isometric, hence we request this from now on. Note: Vg (f ) is a complex-valued function, so we usually look 2 at jVg (f )j, or perhaps better jVg (f )j , which can be viewed as d d a probability distribution over R × Rb if kf k2 = 1 = kgk2. Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem The continuous reconstruction formula Now we can apply a simple abstract principle: Given an isometric embedding T of H1 into H2 the inverse (in the range) is given by ∗ the adjoint operator T : H2 !H1, simply because 8h 2 H1 2 2 ∗ hh; hiH1 = khkH1 = (!) kThkH2 = hTh; ThiH2 = hh; T ThiH1 ; (1) and thus by the polarization principle T ∗T = Id. 2 d In our setting we have (assuming kgk2 = 1) H1 = L (R ) and 2 d d H2 = L (R × Rb ), and T = Vg . It is easy to check that Z ∗ L2 d d Vg (F ) = F (λ)π(λ)g dλ, F 2 (R × Rb ); (2) Rd ×Rbd 2 d understood in the weak sense, i.e. for h 2 L (R ) we expect: Z ∗ hV (F ); hi 2 = F (x) · hπ(λ)g; hi 2 dλ. (3) g L (Rd ) L (Rd ) Rd ×Rbd Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Continuous reconstruction formula II Putting things together we have Z ∗ hf ; hi = hVg (Vg (f )); hi = Vg (f )(λ) · Vg (h)(λ) dλ. (4) Rd ×Rbd A more suggestive presentation uses the symbol gλ := π(λ)g and describes the inversion formula for kgk2 = 1 as: Z 2 d f = hf ; gλi gλ dλ, f 2 L (R ): (5) Rd ×Rbd This is quite analogous to the situation of the Fourier transform Z 2 d f = hf ; χs i χs ds; f 2 L (R ); (6) Rd d with χs (t) = exp(2πihs; ti); t; s 2 R , describing the \pure d frequencies" (plane waves, resp. characters of R ). Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Discretizing the continuous reconstruction formula Note the crucial difference between the classical formula (6) (Fourier inversion) and the new formula formula (5). The building 2 d blocks gλ belong to the Hilbert space L (R ), in contrast to the 2 d characters χs 2= L (R ). Hence finite partial sums cannot 2 d approximate the functions f 2 L (R ) in the Fourier case, but they 2 d can (and in fact do) approximate f in the L (R )-sense. The continuous reconstruction formula suggests that sufficiently fine (and extended) Riemannian-sum-type expressions approximate f . This is a valid view-point, at least for nice windows g (any Schwartz function, or any classical summability kernel is OK: see [F. Weisz] Inversion of the short-time Fourier transform using Riemannian sums for example [7]). Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem S d S 0 d Modulation spaces, in particular 0(R ) and 0 (R ) The reconstruction of f from its STFT (Short-time Fourier Transform) suggests that at least for \good windows" g one can control the smoothness (and/or decay) of a function or distribution by controlling the decay of Vg (f ) in the frequency resp. the time direction. A polynomial weight depending on the frequency variable only can be used to describe Sobolev spaces, and (weighted) mixed-norm conditions can be used to define the (now classical) modulation s d spaces M ( ); k · kMs . p;q R p;q We will put particular emphasis on the modulation spaces d 1;1 1 S0(R ) = M = M , characterized by the membership of 1 2d 0 d 1;1 1 Vg (f ) 2 L (R ) and S0 (R ) = M = M , with uniform 0 d convergence describing norm convergence in S0 (R ), while pointwise convergence corresponds to the w ∗-convergence in 0 d S0 (R ). Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Banach Module Terminology Definition A Banach space (B; k · kB ) is a Banach module over a Banach algebra (A; ·; k · kA) if one has a bilinear mapping (a; b) 7! a • b, from A × B into B bilinear and associative, such that ka • bkB ≤ kakAkbkB 8 a 2 A; b 2 B; (7) a1 • (a2 • b) = (a1 · a2) • b 8a1; a2 2 A; b 2 B: (8) Definition A Banach space (B; k · kB ) is a Banach ideal in (or within, or of) a Banach algebra (A; ·; k · kA) if (B; k · kB ) is continuously embedded into (A; ·; k · kA), and if in addition (7) is valid with respect to the internal multiplication inherited from A. Hans G. Feichtinger FOURIER STANDARD SPACES and the Kernel Theorem Wendel's Theorem Theorem 1 1 1 The space of HL1 (L ; L ) all bounded linear operators on L (G) which commute with translations (or equivalently: with convolutions) is naturally and isometrically identified with M ( b(G); k · kMb ).
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