By Hartmut Führ

ISBN-10: 3540242597

ISBN-13: 9783540242598

This quantity encompasses a systematic dialogue of wavelet-type inversion formulae in keeping with workforce representations, and their shut connection to the Plancherel formulation for in the neighborhood compact teams. the relationship is verified via the dialogue of a toy instance, after which hired for 2 reasons: Mathematically, it serves as a strong software, yielding life effects and standards for inversion formulae which generalize some of the recognized effects. additionally, the relationship offers the start line for a – quite self-contained – exposition of Plancherel thought. for this reason, the ebook is additionally learn as a problem-driven creation to the Plancherel formula.

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**Additional resources for Abstract Harmonic Analysis of Continuous Wavelet Transforms**

**Example text**

In the following, we endow G with left Haar measure. Associate to η ∈ Hπ the orbit (ηx )x∈G = (π(x)η)x∈G . 7(b). (a) η is called admissible iﬀ (π(x)η)x∈G is admissible. (b) If η is admissible, then Vη : Hπ → L2 (G) is called (generalized) continuous wavelet transform. (c) η is called a bounded vector if Vη : Hπ → L2 (G) is bounded on Hπ . We note in passing that η is cyclic iﬀ Vη , this time viewed as an operator Hπ → Cb (G), is injective: Indeed, Vη ϕ = 0 iﬀ ϕ⊥π(G)η, and that is equivalent to the fact that ϕ is orthogonal to the subspace spanned by π(G)η.

Let G be unimodular. Then λG has an admissible vector iﬀ G is discrete. Proof. First, if G is discrete, then the indicator function of {eG }, where eG is the neutral element of G, is admissible: The associated wavelet transform is the identity operator. 40 L2 (G) consists of bounded continuous functions, in particular L2 (G) ⊂ L∞ (G). In order to show that this implies discreteness of G, we ﬁrst show that for G nondiscrete there exist measurable sets of arbitrarily small, positive measure. , := inf{|A| : A ⊂ G Borel , |A| > 0} > 0 .

Is cyclic. Moreover, every η ∈ L2 (R) is admissible up to normalization; more precisely, iﬀ η = 1. 25: G is unimodular, hence the formal dimension operator is a scalar multiple of the identity. In addition, we have established by elementary calculation that the scalar equals one. Since the torus acts by multiplication, we have |Vf (p, q, z)| = |Vf (p, q, 1)|, for all z ∈ T. Hence the map Wf : g → (Vf g)|R2 ×{1} is isometric as well. Wf is the windowed Fourier transform associated to the window f .

### Abstract Harmonic Analysis of Continuous Wavelet Transforms by Hartmut Führ

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