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Multiresolution approximations and wavelet orthonormal bases of $$L^ 2(\mathbb R)$$. (English) Zbl 0686.42018
Summary: A multiresolution approximation is a sequence of embedded vector spaces $$(V_ j)_{j\in \mathbb Z}$$ for approximating $$L^ 2(\mathbb R)$$ functions. We study the properties of a multiresolution approximation and prove that it is characterized by a $$2\pi$$-periodic function which is further described. From any multiresolution approximation, we can derive a function $$\psi$$ (x) called a wavelet such that ($$\sqrt{2^ j}\psi (2^ jx- k))_{(k,j)\in \mathbb Z^ 2}$$ is an orthonormal basis of $$L^ 2(\mathbb R)$$. This provides a new approach for understanding and computing wavelet orthonormal bases. Finally, we characterize the asymptotic decay rate of multiresolution approximation errors for functions in a Sobolev space $$H^ s$$.

MSC:
 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems 41A30 Approximation by other special function classes
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