Localization principle for wavelet expansions (Q2781641)

Consider an orthogonal wavelet basis \( \{ 2^{j/2} \psi(2^j \cdot - n) : j, n \in {\mathbb{Z}} \} \) of \( L^2 (\mathbb{R})\) constructed via a multiresolution analysis generated by an orthonormal scaling function \( \varphi \in L^2 (\mathbb{R})\). NEWLINENEWLINENEWLINEBy periodization one obtains an orthonormal basis of \( L_2(\mathbb{T})\), where \({\mathbb{T}} \) is the unit circle. Then the convergence rate of the wavelet expansion of a function \( f \in L_2 ({\mathbb{T}}) \) at a point \( x_0 \) with \( f(x) = 0 \) in a neighborhood of \(x_0 \) depends on the decay of the scaling function \( \varphi \) and the wavelet function \( \psi \) at infinity. An example shows that the given estimate is sharp.NEWLINENEWLINEFor the entire collection see [Zbl 0971.00015].