Non Stationary Multiresolution Analysis

An orthonormal basis of wavelets of $L^2(\R)$ is an orthonormal basis of $L^2(\R)$ of type
\[
\psi_{j,k}=2^{j/2}\psi(2^j\cdot-k),\quad j,k\in\Z.
\]
A classical method to obtain such bases consists in constructing a multiresolution analysis. When the mother wavelet $\psi$ depends on the scale (i.e. the index $j$), a non stationary version of multiresolution analysis is then used. It is for example the case in the general context of Sobolev spaces. We generalize different characterizations in the standard theory of wavelets to the case of multi-scales wavelets and non stationary multiresolution analyses.