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30 Chapter 2. The Fundamental Property of Wavelets In summary we have f ∈ Wj ⇒ f ˜j ⇒ f f ∈W H s (R) H s (R) ≃ 2js f ≃2 js f Ls (R) , Ls (R) , ˜ R], s ∈ [−R, ˜ s ∈ [−R, R]. 6. 21) is verified by all functions whose Fourier transform is supported in (−∞, −2J ] ∪ [2J , ∞). Such inequalities are inherently bound to the frequency localisation of the functions considered, or, to put it in a different way, to their more or less oscillatory behaviour. Saying that a function is “low frequency” means that such function does not oscillate too much.

5) hold. 7a) holds if and only ˜ we have if for all polynomials p of degree d ≤ M p= p, ϕ˜j,k ϕj,k . 6a) holds if and only if for all polynomials p of degree d ≤ M we have p, ϕj,k ϕ˜j,k . 5). Proof. 13) (the reverse being straightforward). ˜ , and let I =]a, b[⊂ R Let p be a polynomial of degree lower or equal than M ˜ j , a − (L + L)/2 ˜ j [. Consider the be any bounded interval. Let Iˆj =]a − (L + L)/2 L2 (R)-function p˜ coinciding with p in Iˆj and vanishing in L2 (R) \ Iˆj . It is not difficult to realize that, for m ≥ j, if (m, k) is such that supp ψm,k ∩ I = ∅, then ¯ ¯ supp ψ˜m,k ⊆ Iˆj ; analogously if supp ϕj,k ∩ I = ∅, then supp ϕ˜j,k ⊆ Iˆj .

Vectors u of wavelet coefficients of a discrete function. 2. These matrices and vectors are not directly maniable. However, thanks to the properties of wavelets it is in general possible to replace the infinite sum by a finite one without substantially changing the resulting method. For the sake of simplicity let us concentrate on the case of Ω a bounded domain, so that for any fixed level j the cardinality of Λj is finite. 3. Heuristically, the argument that we have in mind is that if a discrete function satisfies an inverse inequality (∼ it is “low frequency”), then the levels in the infinite sum corresponding to “high frequency” components will be negligible and then the infinite sum 52 Chapter 3.