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  "body": "\n## Definition\n\nA *multiresolution analysis* of $L^2(\\R)$ (see [[Lp space|SASP14MA1Z]]) is a sequence $(V_j)_{j \\in \\Z}$ of closed linear subspaces of $L^2(\\R)$ with these properties.\n\n1. Nesting: $V_j \\subseteq V_{j+1}$ for all $j \\in \\Z$.\n2. Scaling: for all $j \\in \\Z$ and $f \\in L^2(\\R)$, $f \\in V_j$ if and only if the function $x \\mapsto f(2x)$ lies in $V_{j+1}$.\n3. Separation and density: $\\bigcap_{j \\in \\Z} V_j = \\set{0}$, and $\\bigcup_{j \\in \\Z} V_j$ is [[dense|0BSJSX7ME2]] in $L^2(\\R)$.\n4. Scaling function: there is $\\varphi \\in V_0$ such that the translates $(\\varphi(\\cdot - k))_{k \\in \\Z}$ form an [[orthonormal basis|0S9GGTR0EX]] of $V_0$.\n\n## Remarks\n\nSome authors only ask in clause 4 that the translates of $\\varphi$ form a Riesz basis of $V_0$; an orthonormalization step then yields a function as in clause 4.\nClauses 1 to 3 imply that $f \\in V_0$ if and only if $f(2^j \\cdot) \\in V_j$.\nLet $W_j$ be the orthogonal complement of $V_j$ in $V_{j+1}$.\nThen $L^2(\\R)$ is the orthogonal direct sum of all $W_j$.\nThere is an orthonormal [[wavelet|4PZ35C0GV3]] $\\psi$ whose integer translates form an orthonormal basis of $W_0$.\n\nExample: let $V_j$ be the set of $f \\in L^2(\\R)$ that are constant on each interval $[k 2^{-j}, (k+1) 2^{-j})$, $k \\in \\Z$.\nNesting and scaling are clear, and step functions on dyadic intervals are dense in $L^2(\\R)$.\nA function in every $V_j$ is constant on $[0, 2^{-j})$ and on $[-2^{-j}, 0)$ for all $j \\le 0$, hence constant on $[0, \\infty)$ and on $(-\\infty, 0)$, hence $0$ since it is square-integrable.\nWith $\\varphi = 1_{[0, 1)}$, clause 4 holds; this is the Haar multiresolution analysis.\n\nNon-example: $V_j = L^2(\\R)$ for all $j$ satisfies nesting, scaling and density, but $\\bigcap_j V_j = L^2(\\R) \\ne \\set{0}$, so separation fails.\n",
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