By Charles K. Chui (Auth.)

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**Additional info for Wavelets. A Tutorial in Theory and Applications**

**Example text**

10) 32 P. Heller et al. by L(ao, a i , . . , a m ^_i) := (a m , a m + i , . . , Û ^ - I , 0 , . . 11) R(a0, a i , . . , a m p _i) := ( 0 , . . , 0, a0, au . . 12) and 4 the left-shift and right-shift operators by m units . We have the usual inner product on F m g , and we see that the adjoint L* of L is precisely R, and conversely. , L9 = R9 = 0. ,m-l, = 0,... ,g - 1,s = 0 , . . , m - 1, = 0 , . . , g - 1, s = 0 , . . , m - 1. 7 = 0 , . . , 0 - 1 . 14) given by (ao,... ,a mp _ 9 _i)i—>(ao,...

Since φ(ί) G Vi C T\ where V\ is part of the L 2 (R) multiresolution analysis, it follows that all partial sums Σ an

(2ί - j) n j = Έ ( Σ a»cJ-2" ) ^ψ^ - i) 3 Clearly if an is of polynomial growth say, 0(|n| fc ), so is a'n. It can also be shown that To is closed in the sense of S'r. This follows from the fact that / m —* 0 in S'r is equivalent to fm = Dr[(l + t2)kd^] and % m | -> 0, (see [26]).

Namely, we have m—1 m—1 / , K K — 2 ^ 2 - / Ömife+ramfc+r r=0 r=0 fcGZ = ΣΧ α » n Corollary 11. 11) is a wavelet matrix whose characteristic Haar matrix is the canonical Haar matrix. The above corollary is an example of the fact that there is a left-action of the unitary group U(m — 1) on the wavelet matrices WM(ra, #; C) given by i/GU(m-l),aeWM(m,5;C)H> f j jja. 12) Corollary 12. 8) is equivariant2 with respect to the left actions of U(ra - 1) on WM(ra, g\ C) and H(ra; C). 3. Tensor products of Haar wavelets matrices Theorem 13.