Multiresolution Analysis

Fourier Analysis · Axiom Academy

LESSON Multiresolution Analysis Understanding the mathematical framework that makes wavelets possible 1. Nested Approximation Spaces An MRA is built on a sequence of nested subspaces of L²(R). Each space V j represents approximations at scale 2 j . As j increases, the spaces contain more detailed functions. The key property is that these spaces nest inside each other, like Russian dolls: Nested: V j ⊂ V j+1 for all j ∈ Z Density: Union of all V j is dense in L²(R) Completeness: Intersection of all V j contains only the zero function Each space V j is generated by translations and dilations of a single function φ(t), called the scaling function or father wavelet. The scaling function satisfies a special property: it can be expressed as a weighted sum of its own dilated and shifted versions. This is called the refinement equation . 3. Wavelet Spaces: Capturing the Details Since V j ⊂ V j+1 , there are functions in V j+1 that are not in V j . These "new" functions at each scale form the wavelet space W j . W j represents the detail or difference between approximations at consecutive scales: The fundamental MRA decomposition shows that any approximation at scale j+1 can be split into an approximation at scale j plus the detail at scale j: 5. Constructing the Wavelet Function The wavelet function ψ(t), also called the mother wavelet , generates the wavelet spaces W j just as φ(t) generates V j .

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