By Patton, C.C. and Campbell, J.M
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5 is valid. 12) has the following limit: ϕ M ∗ − j ξ ϕ M ∗ − j ξ f (ξ ) g (ξ ) dξ = ϕ(0)ϕ(0) f, g , lim j→+∞ Rd where the limit and the integral can be interchanged by the Lebesgue’s dominated convergence theorem. The second summand can be estimated as follows: |R j | ≤ ϕ ∞ ϕ f (ξ ) g ξ + M ∗ j l ∞ Rd dξ. , lim R j = 0. 5 in this case. Now, we assume that f ∈ L 2 (Rd ) and g is such that g is continuous and compactly supported. Then, f can be approximated in L 2 -norm by f such that f is compactly supported and continuous.
2 Let ϕ, ϕ ∈ L 2 (Rd ). 6) for almost all ξ ∈ Rd . 7) 22 2 MRA-Based Wavelet Bases and Frames for almost all ξ ∈ Rd . 1, item (v). 3 Suppose f, ϕ, ϕ ∈ L 2 (Rd ), such that [ϕ, ϕ] ∈ L ∞ (Rd ) and [ϕ, ϕ] ∈ L ∞ (Rd ), j ∈ Z. 9) 2 and B = [ϕ, ϕ] f, ϕ jk ϕ jk ≤ BB f ∞ , and = [ f (M ∗ j ·), ϕ](M ∗− j ·)ϕ(M ∗− j ·). 10) k∈Zd Proof Note that ϕ j0 2 = ϕ 2 . 7 together with f, ϕ jk = f − j,0 , ϕ0k , we get 2 f, ϕ jk ϕ0k k∈Zd =B 2 | f − j,0 , ϕ0k |2 ≤ B B f − j,0 2 2 = BB f 2 2. 5) and f, ϕ jk ϕ jk (ξ ) = F k∈Zd f, ϕ jk e2πi(k,M ∗− j k∈Zd [ f (M ∗ j ·), ϕ](M ∗− j ξ ) ϕ(M ∗− j ξ ).
5 Matrix Extension Problem 45 If the refinable masks m 0 , m 0 are smooth, then the wavelet masks constructed by the above algorithm are not necessarily smooth (or even continuous). Let us show that in the case d < 2m − 1, smoothness can be preserved. Suppose that the functions μ0k , μ0k have continuous partial derivatives of order r , r ≥ 1. Obviously, the functions μ0k defined above are smooth. The smooth mapping (μ00 , . . , μ0,m−1 ) takes the d-dimensional torus Td to a unit sphere S in the complex m-dimensional space Cm .