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  "body": "\n## Definition\n\nLet $f \\colon \\R \\to \\C$ be Lebesgue [[integrable|X7GTF3D98F]].\nLet $\\operatorname{cas}(x) = \\cos x + \\sin x$.\nThe *Hartley transform* of $f$ is the function $\\mathcal{H} f \\colon \\R \\to \\C$ given by\n$$(\\mathcal{H} f)(\\omega) = \\frac{1}{\\sqrt{2\\pi}} \\int_\\R f(t)\\, \\operatorname{cas}(\\omega t)\\, dt.$$\n\n## Remarks\n\nLet $F(\\omega) = \\frac{1}{\\sqrt{2\\pi}} \\int_\\R f(t) e^{-i\\omega t}\\, dt$ be the unitary [[Fourier transform|DPVM34ADZV]] with angular frequency.\nFor real $f$, $\\operatorname{Re} F(\\omega) = \\frac{1}{\\sqrt{2\\pi}} \\int f(t) \\cos(\\omega t)\\, dt$ and $\\operatorname{Im} F(\\omega) = -\\frac{1}{\\sqrt{2\\pi}} \\int f(t) \\sin(\\omega t)\\, dt$, so $\\mathcal{H} f = \\operatorname{Re} F - \\operatorname{Im} F$.\nIn general $\\mathcal{H} f(\\omega) = \\frac{1 + i}{2} F(\\omega) + \\frac{1 - i}{2} F(-\\omega)$.\nFrom this formula and the Fourier inversion theorem, $\\mathcal{H}(\\mathcal{H} f) = f$ when $f$ and $\\mathcal{H} f$ are integrable and $f$ is continuous.\nAlso $\\mathcal{H}$ extends to a unitary operator on $L^2(\\R)$ with $\\mathcal{H}^2 = \\id$.\n\nExample: $f(t) = e^{-t^2/2}$.\nSince $f$ is even, $\\int f(t) \\sin(\\omega t)\\, dt = 0$, and $\\frac{1}{\\sqrt{2\\pi}} \\int e^{-t^2/2} \\cos(\\omega t)\\, dt = e^{-\\omega^2/2}$.\nSo $\\mathcal{H} f(\\omega) = e^{-\\omega^2/2}$.\n\nNon-example: $f(t) = 1$ is not integrable on $\\R$, and $\\int_\\R \\operatorname{cas}(\\omega t)\\, dt$ does not converge for any $\\omega$.\n",
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