multiresolutionanalyyysin

Multiresolutionanalyyysin, often written as multiresolution analysis (MRA) in the literature, is a framework in mathematical analysis and signal processing for representing functions or signals at multiple scales. It provides a structured way to decompose data into coarse approximations and fine details.

An MRA consists of a sequence of closed subspaces {V_j} of the Hilbert space L^2(R) that are nested (V_j ⊂ V_{j+1}) and satisfy two limit conditions: the intersection over all j is {0}, and the union over all j is dense in L^2(R). There exists a scaling function phi such that V_0 = span{phi(x - k): k ∈ Z}, and for each j, V_j = span{2^{j/2} phi(2^j x - k): k ∈ Z}. The scaling property means that dilating by a factor of 2 maps V_j into V_{j+1}.

The orthogonal complement W_j of V_j in V_{j+1} captures the details added at the next coarser scale, and V_{j+1} = V_j ⊕ W_j. A central result is that there exists a wavelet function psi that generates all details across scales: the family {2^{j/2} psi(2^j x - k): j ∈ Z, k ∈ Z} forms an orthonormal basis for L^2(R) when combined with the scaling subspaces.

Common examples include the Haar MRA, using the simplest scaling function phi = 1_{[0,1)}, and Daubechies wavelets, which provide smooth, compactly supported bases. MRA has wide applications in wavelet theory, enabling efficient representations for tasks such as data compression, denoising, feature extraction, and numerical analysis. The framework was developed in the 1980s by Mallat, Meyer, and others, with Daubechies contributing foundational wavelets.