By Hartmut Führ
This quantity features a systematic dialogue of wavelet-type inversion formulae in response to staff representations, and their shut connection to the Plancherel formulation for in the neighborhood compact teams. the relationship is established through the dialogue of a toy instance, after which hired for 2 reasons: Mathematically, it serves as a robust device, yielding lifestyles effects and standards for inversion formulae which generalize some of the identified effects. furthermore, the relationship offers the start line for a – quite self-contained – exposition of Plancherel idea. accordingly, the booklet is additionally learn as a problem-driven creation to the Plancherel formula.
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Additional info for Abstract Harmonic Analysis of Continuous Wavelet Transforms
55 (1) below. Our further discussion requires some basic and widely known facts about tight frames. 53. Let (ηi )i∈I ⊂ H be a tight frame with frame constant c. (a) If H ⊂ H is a closed subspace and P : H → H is the projection onto H , then (P ηi )i∈I is a tight frame of H with frame constant c. (b) Suppose that c = 1. Then (ηi )i∈I is an ONB iﬀ ηi = 1 for all i ∈ I. (c) If ηi = ηj , for all i, j ∈ I, then ηi 2 ≤ c. (d) (ηi )i∈I is an orthonormal basis iﬀ c = 1 and the coeﬃcient operator is onto.
If H contains a nonzero function whose support has ﬁnite Haar measure, there is no admissible vector for H. This concludes the discussion of the relations between continuous wavelet transforms and λG . Let us summarize the main results: • • • A necessary condition for π to have admissible vectors is that π < λG . For nondiscrete unimodular groups, it is not suﬃcient. Embedding π into λG and making suitable identiﬁcations, we may assume that Hπ = L2 (G) ∗ S, with S a selfadjoint convolution idempotent.
Since Cπ−1 is densely deﬁned, these tensors span a dense subspace of Hπ ⊗Hπ . 15), T is isometric. Hence there exists a unique linear isometry, also denoted by T : Hπ ⊗ Hπ → L2π (G). By part (a), it has dense image, hence T is in fact unitary. We will next show that T is an intertwining operator. 16) gives rise to π(x)Cπ−1 π(x)∗ = ∆G (x)−1/2 Cπ−1 . Then we compute 36 2 Wavelet Transforms and Group Representations T (ϕ ⊗ π(x)η)(y) = ϕ, π(y)Cπ−1 π(x)η = ϕ, ∆G π(y)π(x)Cπ−1 η 1/2 = ∆G (x)1/2 T (ϕ ⊗ π)(yx) = ( (T (ϕ ⊗ π)) (y) .