Mathematical Analysis

Download Abstract Harmonic Analysis of Continuous Wavelet Transforms by Hartmut Führ PDF

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By Hartmut Führ

This quantity features a systematic dialogue of wavelet-type inversion formulae according to staff representations, and their shut connection to the Plancherel formulation for in the neighborhood compact teams. the relationship is tested by way of the dialogue of a toy instance, after which hired for 2 reasons: Mathematically, it serves as a strong software, yielding lifestyles effects and standards for inversion formulae which generalize some of the recognized effects. additionally, the relationship offers the place to begin for a – kind of self-contained – exposition of Plancherel concept. for this reason, the ebook can be learn as a problem-driven creation to the Plancherel formula.

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A discrete subgroup Γ < G is called a lattice if the quotient G/Γ carries a finite invariant measure. If a lattice exists, G is unimodular. 4, we let covol(Γ ) = |A| , which is independent of the choice of A. 2). The existence of a lattice implies that G is unimodular. 58. Let Γ < G, and suppose that there exists a frame of the form π(Γ )ϕ, with ϕ ∈ Hπ . Then there exist η ∈ Hπ such that π(Γ )η is a tight frame. Proof. First note that up to normalization the tight frame property is precisely admissibility for the restriction of π to Γ .

29) i∈I Now assume that G is unimodular. Then Cσ = cσ ×IdHσ , with positive scalars cσ . 27) entails the necessary condition mπ (σ)c−2 σ < ∞ . 30) hold, therefore we have found a characterization of direct sums of discrete series representations with admissible vectors. 28). In the nonunimodular case the situation is much less transparent. However, it turns out that the restrictions actually vanish! To begin with, dim(Hσ ) = ∞ follows from the existence of an unbounded operator Cσ on Hσ . 27). Here the fact that the Cπi are unbounded makes it conceivable that there exist vectors that actually fulfill both conditions.

It is based on the ax + b group, the semidirect product R R . As a set G is given as G = R × R , with group law (b, a)(b , a ) = (b + ab , aa ) . The left Haar measure is db|a|−2 da, which is distinct from the right Haar measure db|a|−1 da. Wavelets arise from the quasi-regular representation π acting on L2 (R) via (π(b, a)f )(x) = |a|−1/2 f x−b a . Again, computing L2 -norms of wavelet coefficients turns out to be an exercise in real Fourier analysis. 4 Discrete Series Representations Vη g 2 2 31 2 | g, π(b, a)η | dµG (b, a) = G | gˆ, (π(b, a)η)∧ | dµG (b, a) 2 = G 2 fˆ(γ)|a|1/2 e2πiγb ηˆ(aγ)dγ = R R R 2 fˆ(γ)e2πiγb ηˆ(aγ)dγ = R Rk R |a|−2 dbda |a|−1 dbda 2 = R R φa (−b) |a|−1 dbda, where φa (γ) = gˆ(γ)ˆ η (aγ).

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