A Function Space Defined by the Wigner Transform and Is Applications

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A Function Space Defined by the Wigner Transform and Is Applications Introduction History Time-Frequency Analysis FT on SOPRd Kernel Theorem Various MATLAB illustrations Spectrograms bibliography Nobel Prize Numerical Harmonic Analysis Group A Function Space defined by the Wigner Transform and is Applications Hans G. Feichtinger, Univ. Vienna & TUM [email protected] www.nuhag.eu Eugene Wigner Institute, Budapest . JOINT presentation with Maurice de Gosson Hans G. Feichtinger, Univ. Vienna & TUM [email protected] Function Spacewww.nuhag.eu defined by the Wigner Transform and is Applications Introduction History Time-Frequency Analysis FT on SOPRd Kernel Theorem Various MATLAB illustrations Spectrograms bibliography Nobel Prize Key aspects of my talk 1 Fourier Analysis is a classical topic 2 Time-Frequency Analysis 3 S L2 S0 d The Banach Gelfand Triple ( 0; ; 0)(R ) 4 Various Typical Applications 5 The Idea of Conceptual Harmonic Analysis Hans G. Feichtinger A Function Space defined by the Wigner Transform and is Applications Introduction History Time-Frequency Analysis FT on SOPRd Kernel Theorem Various MATLAB illustrations Spectrograms bibliography Nobel Prize Starting with my personal background My education at the University of Vienna started as a teacher student in Mathematics and Physics; Soon the \imprecision" in physics combined with the BOURBAKI-style (clear and well structured) introduction into analysis (including Lebesgue integrals and functional analysis) let me go more in the direction of (pure) mathematics; In my third year of studies (1971) my then advisor H. Reiter arrived in Vienna, so I became a \Harmonic Analyst"; After my habilitation (1979) I learned about the connection between Fourier Analysis and signal processing in Heidelberg; Since that time I tried to look out for real world applications and call myself now an application oriented mathematician; One of my recent \hobbies" is to promote what I call Conceptional Harmonic Analysis. Hans G. Feichtinger A Function Space defined by the Wigner Transform and is Applications Introduction History Time-Frequency Analysis FT on SOPRd Kernel Theorem Various MATLAB illustrations Spectrograms bibliography Nobel Prize A personal background story At some point in the early 80th I tried to connect with the applied people at TU Vienna, and (fortunately) I ended up contacting Franz Hlawatsch (Communication Theory Dept., TU Vienna). During our first meeting he explained to me, that he was working on the so-called Pseudo-Wigner distribution, which is a kind of smoothed version of the Wigner distribution, with the idea of reducing the so-called interference terms in a Wigner distribution. I told him that I was studying a certain function space (I called it S d 0(R ); k · kS0 , because it was a special Segal algebra), given by: S d X X 0(R ) = ff j f = cnM!n Txn g0; with jcnj < 1g; (1) n≥1 n≥1 2 where g0(x) = exp(−πjxj ) is the usual Gauss function. Hans G. Feichtinger A Function Space defined by the Wigner Transform and is Applications Introduction History Time-Frequency Analysis FT on SOPRd Kernel Theorem Various MATLAB illustrations Spectrograms bibliography Nobel Prize The relevance for my future work At this point none of us had an idea how close we had been in terms of the setting of our research (which was then going on for almost 20 years)! He had two important questions 1 Do you really need all TF-shifts, D. Gabor has suggested to d use only xn;!n 2 Z ! 2 and: How do you compute the coefficients? In fact, according to the claim made in D. Gabor's paper of 1946 [5] one should expect that an optimally centrated representation of any function, using integer TF-shifts of the Gauss function (achieving equality in the Heisenberg Uncertainty Relation!) should be possible. In fact, he only argued that a TF-lattice of the form aZ × bZ with ab > 1 is not comprehensive enough and ab < 1 produces linear dependencies. Hans G. Feichtinger A Function Space defined by the Wigner Transform and is Applications Introduction History Time-Frequency Analysis FT on SOPRd Kernel Theorem Various MATLAB illustrations Spectrograms bibliography Nobel Prize Eugene Wigner Eugene Paul Wigner (November 17, 1902 to January 1, 1995), was a Hungarian-American theoretical physicist, engineer, and mathematician. Nobel Prize in Physics in 1963 for his contributions to the theory of the atomic nucleus and the elementary particles and symmetry principles. Hans G. Feichtinger A Function Space defined by the Wigner Transform and is Applications Introduction History Time-Frequency Analysis FT on SOPRd Kernel Theorem Various MATLAB illustrations Spectrograms bibliography Nobel Prize Dennis Gabor Dennis Gabor (5 June 1900 to 9 February 1979) was a Hungarian-British electrical engineer and physicist, most notable for inventing holography, for which he later received the 1971 Nobel Prize in Physics. Hans G. Feichtinger A Function Space defined by the Wigner Transform and is Applications Introduction History Time-Frequency Analysis FT on SOPRd Kernel Theorem Various MATLAB illustrations Spectrograms bibliography Nobel Prize S d 0(R ) via the Wigner transform Z d n 1 d o S0(R ) = f 2 L (R ): kM!f ∗ f k1 d! < 1 : (2) R2d Condition (2) is equivalent to the integrability of the Wigner d d functions over phase space R × Rb . 2πi!·t d d Here M!f (t) = e f (t), t 2 R ;! 2 R , is the modulation 1 d operator and ∗ is the usual convolution of L (R )-functions. Any d d non-zero function g 2 S0(R ) defines a norm on S0(R ) via Z kf kS0;g = kE!f ∗ gk1 d!; (3) gb d that turns S0(R ) into a Banach space. These norms are pairwise equivalent and we therefore allow ourselves to simply write k · kS0 without specifying n g (see also [2]). Hans G. Feichtinger A Function Space defined by the Wigner Transform and is Applications Introduction History Time-Frequency Analysis FT on SOPRd Kernel Theorem Various MATLAB illustrations Spectrograms bibliography Nobel Prize A schematic description: all the spaces Tempered Distr. L1 L2 C0 SO' S0 Schw FL1 Hans G. Feichtinger A Function Space defined by the Wigner Transform and is Applications Introduction History Time-Frequency Analysis FT on SOPRd Kernel Theorem Various MATLAB illustrations Spectrograms bibliography Nobel Prize Banach Gelfand Triples: the simplified setting Testfunctions ⊂ Hilbert space ⊂ Distributions, like Q ⊂ R ⊂ C! Hans G. Feichtinger A Function Space defined by the Wigner Transform and is Applications Introduction History Time-Frequency Analysis FT on SOPRd Kernel Theorem Various MATLAB illustrations Spectrograms bibliography Nobel Prize 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 A Function Space defined by the Wigner Transform and is Applications Introduction History Time-Frequency Analysis FT on SOPRd Kernel Theorem Various MATLAB illustrations Spectrograms bibliography Nobel Prize A Typical Musical STFT A typical waterfall melody (Beethoven piano sonata) depictured using the spectrogram, displaying the energy distribution in the TF = time-frequency plan: Hans G. Feichtinger A Function Space defined by the Wigner Transform and is Applications Introduction History Time-Frequency Analysis FT on SOPRd Kernel Theorem Various MATLAB illustrations Spectrograms bibliography Nobel Prize A Banach Space of Test Functions (Fei 1979) 2 d d A function in f 2 L (R ) is in the subspace S0(R ) if for some d non-zero g (called the \window") in the Schwartz space S(R ) ZZ kf kS0 := kVg f kL1 = jVg f (x;!)jdxd! < 1: Rd ×Rbd S d The space 0(R ); k · kS0 is a Banach space, for any fixed, d non-zero g 2 S0(R )), and different windows g define the same d space and equivalent norms. Since S0(R ) contains the Schwartz d space S(R ), any Schwartz function is suitable, but also compactly supported functions having an integrable Fourier transform (such as a trapezoidal or triangular function) are suitable. It is convenient to use the Gaussian as a window. Hans G. Feichtinger A Function Space defined by the Wigner Transform and is Applications Introduction History Time-Frequency Analysis FT on SOPRd Kernel Theorem Various MATLAB illustrations Spectrograms bibliography Nobel Prize M1 S d Basic properties of = 0(R ) Lemma d Let f 2 S0(R ), then the following holds: d d d (1) π(u; η)f 2 S0(R ) for (u; η) 2 R × Rb , and kπ(u; η)f kS0 = kf kS0 . ^ S d ^ (2) f 2 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). Hans G. Feichtinger A Function Space defined by the Wigner Transform and is Applications Introduction History Time-Frequency Analysis FT on SOPRd Kernel Theorem Various MATLAB illustrations Spectrograms bibliography Nobel Prize Various Function Spaces FL1 SINC L2 box S0 L2 FL1 L1 Figure: The usual Lebesgues space, the Fourier algebra, and S d the Segal algebra 0(R ) inside all these spaces Hans G. Feichtinger A Function Space defined by the Wigner Transform and is Applications Introduction History Time-Frequency Analysis FT on SOPRd Kernel Theorem Various MATLAB illustrations Spectrograms bibliography Nobel Prize BANACH GELFAND TRIPLES: a new category Definition A triple, consisting of a Banach space (B; k · kB ), which is densely embedded into some Hilbert space H, which in turn is contained in B0 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.
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