By Abul Hasan Siddiqi

ISBN-10: 0824740971

ISBN-13: 9780824740979

ISBN-10: 0824756622

ISBN-13: 9780824756628

Consultant covers the most up-tp-date analytical and numerical equipment in infinite-dimensional areas, introducing contemporary leads to wavelet research as utilized in partial differential equations and sign and photograph processing. For researchers and practitioners. comprises index and references.

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**Example text**

Dρ ρ,σ j,s i,t L(G) = |G| δ δ δ . (ii)) = |G| σ σ ρ ρ T v ,T v dσ t s j i L(G) |G| δρ,σ δi,s δj,t . 9) = ρ σ ϕi,k (g)ϕk,j (h)ϕt,s (h) h∈G k=1 (by (i)) = |G| ρ δρ,σ δj,s ϕi,t (g). 6. 8 The set ρ {ϕi,j : ρ ∈ G, i, j = 1, 2, . . 37) is an orthogonal system in L(G) which is also complete. 37). 36)). 38) g∈G h∈G = σ (f1 )σ (f2 ). In what follows, we set A(G) = ρ∈G Hom(Wρ , Wρ ). The Fourier transform is the linear map F : L(G) → f → A(G) ρ(f ). 39) ρ∈G In other words, Ff is the direct sum of the linear operators ρ(f ) : Wρ → Wρ , ρ ∈ G.

We now add some specific properties that hold in the case the representations are irreducible. 7 Let ρ and σ be two irreducible representations of G. ρ σ (i) (Orthogonality relations) ϕi,j , ϕs,t ρ σ (ii) ϕi,j ∗ ϕs,t = ρ |G| δ δ ϕ . dρ ρ,σ j,s i,t L(G) = |G| δ δ δ . (ii)) = |G| σ σ ρ ρ T v ,T v dσ t s j i L(G) |G| δρ,σ δi,s δj,t . 9) = ρ σ ϕi,k (g)ϕk,j (h)ϕt,s (h) h∈G k=1 (by (i)) = |G| ρ δρ,σ δj,s ϕi,t (g). 6. 8 The set ρ {ϕi,j : ρ ∈ G, i, j = 1, 2, . . 37) is an orthogonal system in L(G) which is also complete.

Indeed, taking an orthonormal basis w1 , w2 , . . , wdρ for W we have dρ tr(Ru,v ) = dρ Ru,v wj , wj W = j =1 wj , u W v, wj W = v, u W. 14), we have Ru,v = Tθu ,v . 27) does. Since W is irreducible, R = cIW for some c ∈ C. 27). 27) we have tr(R) = = 1 tr[ρ(g)Ru,v ρ(g −1 )] |G| g∈G 1 tr[Ru,v ] |G| g∈G = v, u We deduce that c = 1 dρ v, u W R= W. and therefore 1 v, u dρ W IW . 28). (iii) It is obvious that the map v → Tv is antilinear. We now show that it is also bijective. Suppose that T ∈ HomG (W, L(X)).

### Applied functional analysis: numerical methods, wavelets, image processing by Abul Hasan Siddiqi

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