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  "body": "\n## Definition\n\nLet $\\psi$ and $\\tilde\\psi$ be functions in the [[Lp space|SASP14MA1Z]] $L^2(\\R)$.\nFor $j, k \\in \\Z$ put $\\psi_{j,k}(x) = 2^{j/2}\\psi(2^j x - k)$ and $\\tilde\\psi_{j,k}(x) = 2^{j/2}\\tilde\\psi(2^j x - k)$.\nThe pair $(\\psi, \\tilde\\psi)$ is a *biorthogonal wavelet* pair if:\n\n1. the family $\\set{\\psi_{j,k} : j, k \\in \\Z}$ is a [[Riesz basis|ZYDYD7FB1Z]] of $L^2(\\R)$;\n2. the family $\\set{\\tilde\\psi_{j,k} : j, k \\in \\Z}$ is a Riesz basis of $L^2(\\R)$;\n3. $\\ip{\\psi_{j,k}}{\\tilde\\psi_{j',k'}} = 1$ if $(j,k) = (j',k')$ and $0$ otherwise.\n\nThe function $\\tilde\\psi$ is called the dual wavelet of $\\psi$.\n\n## Remarks\n\nUnder these conditions every $f \\in L^2(\\R)$ has the expansions\n$$f = \\sum_{j,k \\in \\Z} \\ip{f}{\\tilde\\psi_{j,k}} \\, \\psi_{j,k} = \\sum_{j,k \\in \\Z} \\ip{f}{\\psi_{j,k}} \\, \\tilde\\psi_{j,k},$$\nwith convergence in $L^2(\\R)$.\n\nExample: if $\\psi$ is an orthonormal [[wavelet|4PZ35C0GV3]], such as the Haar wavelet, then $(\\psi, \\psi)$ is a biorthogonal pair.\nEach family is an orthonormal basis, hence a Riesz basis, and clause 3 is orthonormality.\n\nNon-example: take $\\psi$ the Haar wavelet and $\\tilde\\psi = 2\\psi$.\nBoth families are Riesz bases, but $\\ip{\\psi_{0,0}}{\\tilde\\psi_{0,0}} = 2$, so clause 3 fails.\n\nIn practice biorthogonal pairs come from two multiresolution analyses with dual scaling functions $\\varphi$ and $\\tilde\\varphi$, and the transform is computed with two pairs of finite filters.\nThe Cohen–Daubechies–Feauveau family contains the 5/3 and 9/7 biorthogonal wavelets used in JPEG 2000.\n\n## Explanations\n\n### L1 human\n\nTwo families of wavelets are used: one to measure a signal and one to rebuild it.\nThey must be dual, which means each measuring function gives $1$ on its own partner and $0$ on every other rebuilding function.\n",
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