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Continuous Wavelet Transformation on Homogeneous Spaces

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Continuous Wavelet Transformation on Homogeneous Spaces Continuous Wavelet Transformation on Homogeneous Spaces Dissertation zur Erlangung des mathematisch-naturwissenschaftlichen Doktorgrades “Doctor rerum naturalium” der Georg-August-Universität zu Göttingen im Promotionsprogramm “Mathematical Sciences (Ph.D)” der Georg-August University School of Science (GAUSS) vorgelegt von Burkhard Blobel aus Freiberg/Sa. Göttingen, 2020 Betreuungsausschuss: Prof. Dr. Dorothea Bahns, Mathematisches Institut, Universität Göttingen Prof. Dr. Thomas Schick, Mathematisches Institut, Universität Göttingen Mitglieder der Prüfungskommission: Referent: Prof. Dr. Dorothea Bahns, Mathematisches Institut, Universität Göttingen Korreferent: Prof. Dr. Thomas Schick, Mathematisches Institut, Universität Göttingen Weitere Mitglieder der Prüfungskommission: Jun.-Prof. Dr. Madeleine Jotz Lean Mathematisches Institut, Universität Göttingen Prof. Dr. Ralf Meyer, Mathematisches Institut, Universität Göttingen Prof. Dr. Gerlind Plonka-Hoch, Institut für Numerische und Angewandte Mathematik, Universität Göttingen Prof. Dr. Dominic Schuhmacher Institut für Mathematische Stochastik, Universität Göttingen Tag der mündlichen Prüfung: 11.12.2020 Contents Notation and convention v 1 Introduction 1 2 Basics 7 2.1 Locally compact groups . .7 2.1.1 Topological groups . .7 2.1.2 Homogeneous spaces of locally compact groups . .9 2.1.3 Measures on locally compact spaces . 10 2.1.4 Haar measure . 13 2.1.5 Function spaces . 15 2.1.6 Quasi-invariant measures on the quotient space . 17 2.1.7 Standard Borel spaces . 22 2.2 Representation theory . 27 2.2.1 Unitary representation . 27 2.2.2 Induced representations . 30 2.2.3 Properties of induced representations . 34 2.3 The dual and the quasi-dual of a locally compact group . 35 2.3.1 Von Neumann algebras . 35 2.3.2 Direct integral of Hilbert spaces . 38 2.3.3 Decomposition of representations . 41 2.3.4 Plancherel decomposition . 44 2.3.5 Remarks on the Plancherel Theorem . 47 2.4 Contractions of Lie algebras, Lie groups, and representation . 48 3 Continuous wavelet transformations and admissibility conditions 51 3.1 Continuous wavelet transformations and generalizations . 51 3.1.1 Classical continuous wavelet transformation . 51 3.1.2 Shearlets and semidirect products . 55 3.1.3 Windowed Fourier transformation and coherent state transfor- mation . 57 3.1.4 Coherent states over coadjoint orbits . 59 3.1.5 Continuous wavelet transformations on manifolds . 60 iv Contents 3.1.6 Continuous diffusion wavelet transformations . 63 3.2 Group-theoretical approach to continuous wavelet transformations . 64 3.2.1 Introduction and definitions . 65 3.2.2 Square-integrability condition . 67 4 New approaches to continuous wavelet transformations 75 4.1 Relation between representations . 76 4.2 Continuous wavelet transformations on unimodular groups . 80 4.3 Continuous wavelet transformations on homogeneous spaces . 81 4.3.1 Semidirect products . 82 4.3.2 Examples . 95 4.3.3 Group extensions . 97 4.4 Continuous wavelet transformations on manifolds . 101 5 Outlook and discussion 105 A Polar decomposition of direct integral operators 109 B Fourier transformation of tempered distributions 113 Notation and convention Notation im ϕ image of a map ϕ ker ϕ kernel of a homomorphism ϕ At transpose of a matrix A A−t inverse transposed matrix charE characteristic function of a measurable set E ess sup E essential supremum of a measurable set, ess sup E = inf b ∈ R {x ∈ E | x > b} is a null set Cc(X) space of compactly supported continuous functions C0(X) space of continuous functions vanishing at infinity Cb(X) space of bounded continuous functions k · k∞ uniform norm (supremum norm) 1 ∈ H identity operator on a Hilbert space H B(V1; V2) space of bounded linear operators from a Banach space V1 to a Banach space V2 B(H) = B(H; H) space of bounded linear operators on a Hilbert space H (defined via functional calculus) B1(H) space of trace-class operators on a Hilbert space H B2(H) space of Hilbert-Schmidt operators on a Hilbert space H U(H) space of unitary operators on H |T | = (T ∗T )1/2 absolute value of an operator T : H → H, kT k1 trace norm of an operator T : H → H kT k2 Hilbert-Schmidt norm of an operator T : H → H vi Notation ∆G modular function of a locally compact group G λG left regular representation of a locally compact group G µG Plancherel measure of a locally compact group G N0 = {0, 1, 2, ···} natural numbers including 0 N = {1, 2, 3, ···} natural numbers without 0 N = N ∪ {∞} countable cardinalities without 0 Convention R e2πihk,xif x x f ∈ L1 n Rn ( ) d Fourier transform of a function (R ), 1 n 2 n Plancherel transform of a function f ∈ L (R ) ∩ L (R ) h·, ·i (complex) scalar product, antilinear in the first argument, linear in the second argument R G f(a) da integration against the left Haar measure of a locally compact group G H ≤ GH is closed subgroup of a locally compact group G Terminology • neighborhood of a point: a subset of a topological space containing the point in its interior (not necessarily open) • action: left action • representation: strongly continuous unitary representation CHAPTER 1 Introduction Wavelet transformation is a tool coming from data analysis. Roughly speaking, it has its origin in analyzing seismic measurements in geophysics and goes back to Goupillaud, Grossmann, and Morlet. In [38], they discussed the problem of reconstructing and resolving underground structures in order to find reservoirs of oil and gas. The data used for that purpose consists of a superposition of seismic waves, which are backscattered by the different layers in the ground and are measured over time. Since information about the thicknesses and impedances of different layers are encoded in frequencies and timings, it is important to keep track of both. The straightforward approach uses windowed or short-time Fourier analysis, which means that the measured signal is decomposed into “elementary wavelets” of the form it/a ψb,a(t) = e ψ(t − b), (1.1) where ψ is a window function, which is typically chosen to be a Gaussian of width T , 2 1 − t ψ(t) = √ e 2T 2 . (1.2) 2πT 2 This method goes back to Gabor [36]. Goupillaud, Grossmann, and Morlet demonstrate that because of the fixed width of the window, the timing resolution for high frequencies drops and results in a loss of information. Instead, they propose to use a family of waves which scale not only in frequency but also in timing. They decompose the signal into wavelets of the form − 1 −1 φb,a(t) = |a| 2 φ(a (t − b)), (1.3) where φ can be chosen, for example, as 2 2 √ − t − t φ(t) = 2 e T 2 − e 2T 2 . (1.4) The different shapes are illustrated in Fig. 1.1. With this approach Goupillaud, Gross- mann, and Morlet managed to improve the quality and resolution of the results. To evaluate the measured signal f, one has to transform it into a function Vφf(b, a) 2 Introduction low frequencies medium frequencies high frequencies WFT Re(ψb,a(t)) Re(ψb,a(t)) Re(ψb,a(t)) low frequencies medium frequencies high frequencies CWT φb,a(t) φb,a(t) φb,a(t) Figure 1.1: Comparison of wavelets used for windowed Fourier transformation (WFT) and continuous wavelet transformation (CWT) at low (a = 3), medium (a = 1), and high frequencies (a = 1/3). ψb,a is defined in eq. (1.1) and eq. (1.2) with T = 2. φb,a is defined in eq. (1.3) and eq. (1.4) with T = 1. such that Z Z 1 f(t) = Vφf(b, a) · φb,a(t) 2 db da, (1.5) R6=0 R a stating that f is a superposition of the functions {φb,a}. In order to do that they noted 2 2 1 that for suitable φ the operator Vφ : L (R) → L (R × R6=0, a2 db da), Z Vφf(b, a) = φb,a(t)f(t) dt, (1.6) R ∗ 2 1 2 is an isometry. Hence, its adjoint operator Vφ : L (R × R6=0, a2 db da) → L (R), Z Z ∗ 1 Vφ F (t) = F (b, a) · φb,a(t) 2 db da, R6=0 R a has the desired property ∗ f = Vφ Vφf and Vφ defined in eq. (1.6) fulfills the relation in eq. (1.5). Goupillaud, Grossmann, and Morlet called the operator Vφ the voice transformation. Today, it is known as (classical) continuous wavelet transformation. In [39], Grossmann, Morlet, and Paul showed that it is no coincidence that Vφ is 3 an isometry. They stated that the map π(b, a): φ 7→ φb,a is an irreducible unitary representation of the affine group of the real line Gaff (R) = R o R6=0 = {(b, a) | a ∈ R6=0, b ∈ R} with the group law (b, a) · (b0, a0) = (ab0 + b, aa0). Because of its action (b, a).x = ax + b, for x ∈ R, Gaff (R) is also known as the ax + b-group. It turns out that the operator Vφ 2 2 is an intertwiner L (R) → L (Gaff (R)) between π and the left regular representation 1 of Gaff (R). The measure a2 db da is nothing but the Haar measure of Gaff (R). This observation opened the field to pure mathematics including representation theory of locally compact groups and abstract harmonic analysis. During the last decades the idea developed and led to the following definition (cf. Führ [33, Chp. 2.3]). Definition 1.1. Let G be a locally compact group and π a (strongly continuous) unitary representation on a complex Hilbert space H. Let ψ ∈ H be a fixed, nonzero vector. (i) Denote by Vψ the possibly unbounded (not even densely defined) operator H → 2 L (G) given by Vψf(a) = hπ(a)ψ, fi.
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