= Multiresolution analysis
{wiki}
A multiresolution analysis organizes a <function> into successively finer approximation spaces. In the usual orthonormal construction on $L^2(\mathbb R)$, these are <closed> <linear subspaces> $(V_j)_{j\in\mathbb Z}$ with
$$
V_j\subseteq V_{j+1},\qquad
\overline{\bigcup_jV_j}=L^2(\mathbb R),\qquad
\bigcap_jV_j=\{0\}.
$$
Rescaling relates the levels: $f(x)\in V_j$ if and only if $f(2x)\in V_{j+1}$. Integer translates of a <scaling function> form an <orthonormal basis> of $V_0$. The detail space is the <orthogonal complement> of $V_j$ inside $V_{j+1}$, giving $V_{j+1}=V_j\oplus W_j$. Coarse information and new details therefore fit together without overlap in squared <L2 norm>.
For the <Haar wavelet>, $V_j$ consists of <square-integrable functions> constant on cells of length $2^{-j}$. Splitting every cell into two halves creates one new contrast per old cell, the <Haar refinement identity>. On the bounded example $[0,1]$, the first three spaces have dimensions $1,2,4$: a whole-interval mean, two half-interval means, and four quarter-interval means. The <Haar projection> diagram displays those three resolutions. This bounded construction starts at level zero; the displayed intersection condition refers to the full family on the <real line>.
The usefulness is a controlled way to increase resolution. Earlier coarse information remains available while new <wavelet> coefficients identify what the finer description adds. The <wavelet approximation complexity> connects this extra resolution to both approximation error and the number of coefficients being estimated. https://arxiv.org/pdf/math/9304214[Gilbert Strang (1993), sections 1 and 3: Haar spaces, refinement and multiresolution]{ref}
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