Multiresolution analysis

Summary

A multiresolution analysis of $L^2(\R)$ is a nested chain of closed subspaces $V_j$, one for each scale $2^{-j}$. The spaces are related by dilation by $2$, and $V_0$ is spanned by the integer translates of one scaling function. Stéphane Mallat and Yves Meyer introduced it as the framework in which orthonormal wavelets are built. The wavelet spaces are the orthogonal complements of $V_j$ in $V_{j+1}$, and the discrete wavelet transform moves between the levels with two filters.