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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.