The Richness of Banach Spaces Within a Rigged Hilbert Space Resp.: Banach Gelfand Triple

Numerical Harmonic Analysis Group

The Richness of Banach Spaces within a Rigged Hilbert Space resp.: Banach Gelfand Triple

Hans G. Feichtinger [email protected] www.nuhag.eu

GFSE2010 Novi Sad, June 5-th 2010 Honoring Stevan Pilipovic 60th anniversary

Hans G. Feichtinger [email protected] www.nuhag.euThe Richness of Banach Spaces within a Rigged Hilbert Space resp.: Banach Gelfand Triple OVERVIEW over this lecture 28 MINUTES

The classical view on the Fourier transform, using 1 d 2 d d 0 d L (R ), L (R ), S(R ), S (R ); Present some reflections concerning the (generalized) Fourier transform; Offer some hints concerning the Banach Gelfand Triple d based on the Segal algebar S0(R ); Define Standard Spaces and show their richness; Indicate a number of constructions within this family of space; Present a number of images of function spaces;

Hans G. Feichtinger The Richness of Banach Spaces within a Rigged Hilbert Space resp.: Banach Gelfand Triple A variety of function spaces

Hans G. Feichtinger The Richness of Banach Spaces within a Rigged Hilbert Space resp.: Banach Gelfand Triple The classical view on the Fourier Transform

Tempered Distr.

L2 C0

Schw FL1

Hans G. Feichtinger The Richness of Banach Spaces within a Rigged Hilbert Space resp.: Banach Gelfand Triple The classical view on the Fourier Transform

Hans G. Feichtinger The Richness of Banach Spaces within a Rigged Hilbert Space resp.: Banach Gelfand Triple How and where do we our calculations

All the actual computations are done in Q, e.g. using finite decimal expressions, think of multiplications of fractions! The real number R have the advantage of being a complete√ metric spaces, this allows us to define numbers such as 2. Still we cannot solve quadratic equations, and by the miracle of adding the imaginary unit (i.e. introduce new objects: pairs of real numbers with a new multiplication!) one has an even more comprehensive field; IMPORTANT: each time one has a natural embedding of the smaller field within the larger object (e.g. convert fractions into periodic infinite decimal expressions);

Hans G. Feichtinger The Richness of Banach Spaces within a Rigged Hilbert Space resp.: Banach Gelfand Triple A Typical Musical STFT

Hans G. Feichtinger The Richness of Banach Spaces within a Rigged Hilbert Space resp.: Banach Gelfand Triple The idea of a “localized Fourier Spectrum”

partition of unity 1 0.8 0.6 0.4 0.2 0

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real/imag part of signal 0.1

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−0.1 Hans G. Feichtinger The Richness of Banach Spaces within a Rigged Hilbert Space resp.: Banach Gelfand Triple −200 −100 0 100 200 The localized Fourier transform (spectrogram)

part 1 part 2 part 3 0.05 0.05 0.05 0 0 0 −0.05 −0.05 −0.05 −200 0 200 −200 0 200 −200 0 200 part 4 part 5 part 6 0.05 0.05 0.05

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−0.05 −0.05 −0.05 −200 0 200 −200 0 200 −200 0 200 part 7 part 8 part 9 0.04 0.1 0.05 0.02 0.05 0 0 0 −0.02 −0.05 −0.04 −0.05 −200 0 200 −200 0 200 −200 0 200

Hans G. Feichtinger The Richness of Banach Spaces within a Rigged Hilbert Space resp.: Banach Gelfand Triple d Deﬁnition of the Segal algebra S0(R ), k · kS0

d A continuous, integrable function f on R belongs to Feichtinger’s d algebra S0(R ), if its short-time Fourier transform Z −2πi ω t d Vg f (x, ω) := f (t) g(t − x) e dt, x, ω ∈ R , R2d is integrable, where g(t) := e−π|t|2 is the Gaussian window. Here, |t| is the Euclidean norm. The S0-norm is given by Z

kf kS0 := |Vg f (x, ω)| dx dω. R2d For various useful characterizations of S0 and its signiﬁcance in time-frequency analysis, d The Segal algebra S0(R ) is also described as the Wiener amalgam space W(FL1, `1), where the local norm is the Fourier algebra norm. In particular, the compactly supported functions in FL1 and d in S0(R ) are the same. Hans G. Feichtinger The Richness of Banach Spaces within a Rigged Hilbert Space resp.: Banach Gelfand Triple Intersection of Weighted L2-spaces with Sobolev

Hans G. Feichtinger The Richness of Banach Spaces within a Rigged Hilbert Space resp.: Banach Gelfand Triple Describing the Fourier Transform

d At the level of S0(R ), k · kS0 the Fourier transform and! it’s inverse are well deﬁned integral transformation, even Poisson’s formula is strictly valid; 2 d At the level of L (R ), k · k2 one can express the fact that F is unitary mapping, preserving orthogonality and “energy”; At the distributional level one can characterize the linear ∞ 0 d mapping which maps “pure frequencies” (in L ⊂ S0 (R )) to the corresponding Dirac measures;

Hans G. Feichtinger The Richness of Banach Spaces within a Rigged Hilbert Space resp.: Banach Gelfand Triple The kernel theorem (despite lack of nuclearity!)

d S0(R ), k · kS0 is a ptw. and convolutive algebra, invariant under the Fourier transform; d The so-called tensor product property of S0(R ), k · kS0 allows to prove a so-called kernel theorem: Every linear d operator from S0(R ), k · kS0 into SORdPN can be uniquely 0 2d represented by a distributional kernel from S0 (R ); The integral operator is regularizing, i.e. mapping ∗ 0 d 0 w −-convergent sequences in (S0 (R ), k · kS0 ) into norm d convergent sequences in S0(R ), k · kS0 if and only if it is in 2d S0(R ). In between these two extremes one has: An operator is a 2 d Hilbert Schmidt operator on L (R ), k · k2 if and only if its 2 2d kernel is in L (R ).

Hans G. Feichtinger The Richness of Banach Spaces within a Rigged Hilbert Space resp.: Banach Gelfand Triple Regularizing Operators

There are lots of regularizing operators: Any product-convolution operator (with a pointwise multiplier d h ∈ SORd and a convolutive kernel g ∈ S0(R ) is also regularizing in the above sense. For f ∈ SORdN the action of Dirac kernel (compression of 1 d ˆ some g ∈ L (R ), k · k1 with (g) = 1 is approaching the identity operator; 1 d Dilation of a function h ∈ FL (R ) (with h(0) = 1) is giving a pointwise-approximate identity by (ordinary) dilation. Finite partial sums of Gabor expansions are another very useful class of regularizing operators tending to the identity 0 d ∗ operator (on S0 (R ) only in the w −topology).

Hans G. Feichtinger The Richness of Banach Spaces within a Rigged Hilbert Space resp.: Banach Gelfand Triple Relative Completion and Minimal Space

d S0(R ), k · kS0 is the smallest Banach space of functions which is isometrically invariant under time-frequency shifts (and containing at least on non-zero Schwartz function); 0 d 0 (S0 (R ), k · kS0 ) is thus correspondingly the biggest space of (tempered) distributions which is isometrically invariant under time-frequency shifts We call [restricted] standard spaces Banach space B k · kB d 0 d betwenn S0(R ) and S0 (R ), which are also pointwise 1 d modules over FL (R ), k · kFL1 and convolutive modules 1 d under L (R ), k · k1 .

Hans G. Feichtinger The Richness of Banach Spaces within a Rigged Hilbert Space resp.: Banach Gelfand Triple Relative Completion and Minimal Space

For every standard space the closure of the test-function space d S0(R ) in B k · kB is again a standard space; If this is the B itself it is called “minimal”; It’s dual space is in the same class as well; For every standard space B k · kB there is a relative 0 d completion of B k · kB within S0 (R ): on denotes by Be the Banach space of all limits of w ∗-convergent, bounded sequences from B k · kB , with the inﬁmum over all admissible norms of approximating sequences as natural norm. This is the largest space containing B k · kB as subspace with the same norm every such “‘maximal” is the dual space of some minimal space.

Hans G. Feichtinger The Richness of Banach Spaces within a Rigged Hilbert Space resp.: Banach Gelfand Triple Relative Completion and Minimal Space

A standard Banach space B k · kB is reﬂexive if and only it is minimal as well as maximal, and its dual has the same property! Given a standard space its Fourier image is a standard space as well; for every pair of standard spaces the set of all operator kernels mapping one into the other is again a standard space (on 2d R ); For any two standard spaces the set of pointwise multipliers from one into the other is either trivial or a standard space as well; same for convolution kernels; for any standard space the Wiener amalgam space W(B, `p) is again a standard space;

Hans G. Feichtinger The Richness of Banach Spaces within a Rigged Hilbert Space resp.: Banach Gelfand Triple The setting of Ultradistributions

Hans G. Feichtinger The Richness of Banach Spaces within a Rigged Hilbert Space resp.: Banach Gelfand Triple For pseudo-diﬀerential operators: Shubin Classes

Hans G. Feichtinger The Richness of Banach Spaces within a Rigged Hilbert Space resp.: Banach Gelfand Triple BIBLIOGRAPHY

W. Braun and H. G. Feichtinger. Banach spaces of distributions having two module structures. J. Funct. Anal., 51:174–212, 1983. H. Feichtinger. Banach Gelfand triples for applications in physics and engineering. volume 1146 of AIP Conf. Proc., pages 189–228. Amer. Inst. Phys., 2009. H. G. Feichtinger. On a new Segal algebra. Monatsh. Math., 92:269–289, 1981. H. G. Feichtinger and T. Strohmer. Gabor Analysis and Algorithms. Theory and Applications. Birkh¨auser,Boston, 1998.

Hans G. Feichtinger The Richness of Banach Spaces within a Rigged Hilbert Space resp.: Banach Gelfand Triple The END

All the BEST to Stevan!! Thanks for going ahead. and Thank you for your attention

Material will become downloadable at www.nuhag.eu: DB+tools >> talks resp. the conference Web-page.

Hans G. Feichtinger The Richness of Banach Spaces within a Rigged Hilbert Space resp.: Banach Gelfand Triple