Multiresolutionanalyyysi

Multiresolutionanalyyysi is a mathematical framework used in harmonic analysis and signal processing to decompose functions into components at different scales; in English-language literature it is typically called multiresolution analysis (MRA).

An MRA consists of a nested sequence of closed subspaces V_j of L^2(R) such that V_j ⊂ V_{j+1}, the intersection over all j is {0}, and the union over all j is dense in L^2(R). There exists a scaling function φ that generates V_0 by integer translates, and the dilation by a fixed factor (commonly 2) maps V_j into V_{j+1}.

There exist detail spaces W_j defined by V_{j+1} = V_j ⊕ W_j, and the mother wavelet ψ spans W_0, with ψ dilated and translated to form a basis for L^2(R) across scales.

In practice, multiresolution analysis leads to wavelet representations and efficient algorithms for the discrete wavelet transform through filter banks with low-pass and high-pass filters satisfying quadrature mirror filter relations.

Historically, MRA was developed in the 1980s and 1990s by Stéphane Mallat and Yves Meyer, providing a rigorous foundation for wavelet theory and its applications.

Beyond one dimension, multiresolution analysis extends to higher dimensions and various domains, enabling scalable representations in image processing, geophysics, and data analysis.