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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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