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  "body": "\n## Definition\n\nLet $\\psi \\in L^2(\\R)$ (see [[Lp space|SASP14MA1Z]]), and let $\\hat\\psi(\\xi) = \\int_\\R \\psi(t) e^{-2\\pi i t \\xi} \\, dt$ be its [[Fourier transform|DPVM34ADZV]].\nThe function $\\psi$ satisfies the *admissibility condition*, and is called an *admissible wavelet*, if\n$$C_\\psi = \\int_\\R \\frac{\\abs{\\hat\\psi(\\xi)}^2}{\\abs{\\xi}} \\, d\\xi \u003c \\infty.$$\n\n## Remarks\n\nIf $\\psi$ is also in $L^1(\\R)$, then $\\hat\\psi$ is continuous, and admissibility implies $\\hat\\psi(0) = \\int_\\R \\psi(t) \\, dt = 0$.\nConversely, if $\\int_\\R (1 + \\abs{t}) \\abs{\\psi(t)} \\, dt \u003c \\infty$ and $\\int_\\R \\psi = 0$, then $\\hat\\psi$ is Lipschitz with $\\hat\\psi(0) = 0$, and $\\psi$ is admissible.\nThe constant $C_\\psi$ is the factor in Calderón's identity for the continuous wavelet transform.\n\nExample: the Mexican hat $\\psi(t) = (1 - t^2) e^{-t^2/2}$ is admissible.\nIt decays fast and $\\int_\\R \\psi = \\int_\\R e^{-t^2/2} \\, dt - \\int_\\R t^2 e^{-t^2/2} \\, dt = \\sqrt{2\\pi} - \\sqrt{2\\pi} = 0$, so the converse criterion applies.\n\nNon-example: the Gaussian $\\psi(t) = e^{-\\pi t^2}$ has $\\hat\\psi(\\xi) = e^{-\\pi \\xi^2}$.\nNear $\\xi = 0$ the integrand behaves like $1/\\abs{\\xi}$, which is not integrable, so $C_\\psi = \\infty$.\n",
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