Abstract Harmonic Analysis of Continuous Wavelet Transforms - download pdf or read online

By Hartmut Führ

ISBN-10: 3540242597

ISBN-13: 9783540242598

This quantity includes 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 verified 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 a number of the identified effects. in addition, the relationship presents the start line for a – quite self-contained – exposition of Plancherel conception. as a result, the publication is additionally learn as a problem-driven creation to the Plancherel formula.

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Suppose that f ∈ L2 (G). (a) Vf : L2 (G) → L2 (G) is a closed operator with domain dom(Vf ) = {g ∈ L2 (G) : g ∗ f ∗ ∈ L2 (G)} , and acts by Vf g = g ∗ f ∗ . The subspace dom(Vf ) is invariant under left translations. (b) If f ∆−1/2 ∈ L1 (G), then f is a bounded vector, with Vf ≤ f ∆−1/2 1 . This holds in particular when f has compact support. (c) If f ∗ ∈ L2 (G) then L1 (G) ∩ L2 (G) ⊂ dom(Vf ). (d) Suppose that f ∗ ∈ L2 (G). Then Vf∗ ⊂ Vf ∗ . If one of the operators is bounded, so is the other, and they coincide.

25) Proof. 23. 25(c), Cπi ηi = 1 is the admissibility condition on ηi . 15) shows that whenever πi πj , Im(Vηi )⊥Im(Vηi ) ⇐⇒ Cπj Si,j ηi , Cπj ηj = 0 . 24 yields Im(Vηi )⊥Im(Vηi ) for arbitrary vectors ηi and ηj , whenever πi πj . Hence the proof is finished. 22. Here we only consider the case of direct sums of discrete series representations. Some of the phenomena encountered in the general case can be already examined in this simpler setting, in particular the striking difference between unimodular and nonunimodular groups and the role of the formal dimension operators in this context.

Let S ∈ L2 (G) be a selfadjoint convolution idempotent, then for all f ∈ H = L2 (G) ∗ S we have f ∞ ≤ f 2 S 2 . Proof. This follows from the Cauchy-Schwarz inequality: |f (x)| = |(f ∗ S ∗ )(x)| = | f, λG (x)S | ≤ f 2 S 2 . The following proposition gives rise to a somewhat subtle distinction between unimodular and nonunimodular groups: In the unimodular case, any invariant subspace of L2 (G) which has admissible vectors possesses one in the form of a convolution idempotent. 43 below. 40. Suppose that H ⊂ L2 (G) is closed and leftinvariant.

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Abstract Harmonic Analysis of Continuous Wavelet Transforms by Hartmut Führ

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