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. L1 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 The classical view on the Fourier Transform 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 completep 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 50 100 150 200 250 300 350 400 450 500 real/imag part of signal 0.1 0.05 0 −0.05 −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 0 0 0 −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 ! t d Vg f (x;!) := f (t) g(t − x) e dt; x;! 2 R ; R2d is integrable, where g(t) := e−πjtj2 is the Gaussian window. Here, jtj is the Euclidean norm. The S0-norm is given by Z kf kS0 := jVg f (x;!)j dx d!: 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 1 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 2 SORd and a convolutive kernel g 2 S0(R ) is also regularizing in the above sense. For f 2 SORdN the action of Dirac kernel (compression of 1 d ^ some g 2 L (R ); k · k1 with (g) = 1 is approaching the identity operator; 1 d Dilation of a function h 2 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.