Abstract Harmonic Analysis of Continuous Wavelet Transforms by Hartmut Führ

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why study in uk personal statement This quantity includes a systematic dialogue of wavelet-type inversion formulae according to workforce representations, and their shut connection to the Plancherel formulation for in the neighborhood compact teams. the relationship is verified by means of 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. in addition, the relationship presents the place to begin for a – quite self-contained – exposition of Plancherel thought. as a result, the ebook is usually learn as a problem-driven creation to the Plancherel formula.

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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) . This shows that T intertwines 1 ⊗ π with G ; the left half is obvious. Now part (c) is obtained by applying T to the ONB (ηi ⊗ ηj )i,j∈I . 24. Observe that an ONB as in part (c) of the theorem always exists, since dom(Cπ−1 ) is dense; simply apply Gram-Schmidt orthonormalization.

Proof. Let an ONB (ηi )i∈I ⊂ dom(Cπ−1 ) of Hπ be given. Then by part (c) of the previous theorem, we can compute the norm of Pπ (f ) as Pπ (f ) 2 2 = f, VCπ−1 ηj ηi i,j∈I f (x) ηi , π(x)Cπ−1 ηj dx = i,j∈I 2 G | π(f )Cπ−1 ηj , ηi |2 , = i,j∈I where the last equation used the definition of the weak operator integral. But the last term is just the Hilbert-Schmidt norm of π(f )Cπ−1 . Let us now give a few examples for which the discrete series approach cannot work. 4 Discrete Series Representations 37 representation is in the discrete series: Wavelet coefficients are bounded functions and the Haar measure is finite, hence every wavelet coefficient is trivially in L2 .

Since Vη η = 0, the constant is nonzero, and thus η is admissible up to normalization. The construction of the operators Cπ requires additional tools from functional analysis. 15) for the special case that ϕ = ϕ . The domain of this form is D × D, where D is the space of vectors η which are admissible up to normalization. Note that D is dense, being nonzero and invariant. 28 2 Wavelet Transforms and Group Representations Recalling from linear algebra the representation theorem establishing a close connection between quadratic forms and symmetric matrices, we are looking for a positive selfadjoint operator A such that Bϕ (η, η ) = Aη, η , and then letting Cπ = A1/2 should do the trick.

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