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  "body": "\n## Summary\n\nThe Hartley transform is an integral transform like the [[Fourier transform|DPVM34ADZV]], but with the real kernel $\\cos(\\omega t) + \\sin(\\omega t)$ in place of $e^{-i\\omega t}$.\nIt sends real functions to real functions, and with the symmetric normalization it is its own inverse.\nFor a real function it carries the same information as the Fourier transform.\nFor a real function $f$, the even part of the Hartley transform of $f$ is the real part of the Fourier transform of $f$.\nIts odd part is minus the imaginary part of the Fourier transform of $f$.\nIts discrete version is used as a real-arithmetic alternative to the discrete Fourier transform.\n\n## Explanations\n\n### L1 human\n\nThe Fourier transform describes a real signal by complex numbers.\nThe Hartley transform packs the same information into real numbers, by adding the cosine and the sine parts instead of keeping them apart.\nApplying it twice gives back the signal.\nFor example, the Gaussian $e^{-t^2/2}$ is even, so its sine part vanishes and its Hartley transform is again $e^{-\\omega^2/2}$.\n",
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