A note about non stationary multiresolution analysis

An orthonormal basis of wavelets of $L^2(\mathbb{R})$ is an orthonormal basis of $L^2(\mathbb{R})$ of type
\[
\psi_{j,k}=2^{j/2}\psi(2^j\cdot-k),\quad j,k\in\mathbb{Z}.
\]
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. We generalize different characterizations of orthonormal bases of wavelets to the non stationary case (as main reference for the stationary case, we used results presented in "A First Course of Wavelets" of E. Hernandez and G. Weiss).