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  "body": "\n## Definition\n\nLet $\\psi \\in L^2(\\R)$ be a nonzero function that satisfies the [[admissibility condition|2BBB8NY5ZA]]\n$$C_\\psi = \\int_\\R \\frac{\\abs{\\hat\\psi(\\xi)}^2}{\\abs{\\xi}} \\, d\\xi \u003c \\infty,$$\nwhere $\\hat\\psi(\\xi) = \\int_\\R \\psi(t) e^{-2\\pi i t \\xi} \\, dt$ is the [[Fourier transform|DPVM34ADZV]].\nLet $f \\in L^2(\\R)$ (see [[Lp space|SASP14MA1Z]]).\nLet $a \\ne 0$ (the scale) and $b$ (the position) be real numbers.\nThe *continuous wavelet transform* of $f$ with respect to $\\psi$ is\n$$W_\\psi f(a, b) = \\frac{1}{\\sqrt{\\abs{a}}} \\int_\\R f(t) \\, \\overline{\\psi\\!\\left(\\frac{t - b}{a}\\right)} \\, dt.$$\n\n## Remarks\n\nThe integral converges by the Cauchy–Schwarz inequality, and $\\abs{W_\\psi f(a, b)} \\le \\norm{f}_2 \\norm{\\psi}_2$.\nTaking the Fourier transform in $b$ and using the Plancherel theorem gives Calderón's identity\n$$\\int_{\\R \\setminus \\set{0}} \\int_\\R \\abs{W_\\psi f(a, b)}^2 \\, db \\, \\frac{da}{a^2} = C_\\psi \\norm{f}_2^2.$$\nSo $f$ can be recovered from $W_\\psi f$, in a weak sense, by $f = C_\\psi^{-1} \\int \\int W_\\psi f(a, b) \\, \\abs{a}^{-1/2} \\psi((\\cdot - b)/a) \\, db \\, da / a^2$.\nMany authors take only $a \u003e 0$; then the constant changes and, for real $\\psi$, one obtains $\\tfrac12 C_\\psi$.\n\nExample: the Mexican hat $\\psi(t) = (1 - t^2) e^{-t^2/2}$ has $\\hat\\psi(\\xi)$ equal to a constant times $\\xi^2 e^{-2\\pi^2 \\xi^2}$, so $C_\\psi \u003c \\infty$ and $W_\\psi$ is defined.\n\nNon-example: for the Gaussian $\\psi(t) = e^{-\\pi t^2}$, $\\hat\\psi(0) = 1$, and $C_\\psi$ diverges near $\\xi = 0$.\nThe formula for $W_\\psi f$ still makes sense, but Calderón's identity fails, and this choice is excluded by the definition.\n",
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