In mathematics, the Hartley transform (HT) is an integral transform closely related to the Fourier transform (FT), but which transforms real-valued functions to real-valued functions. It was proposed as an alternative to the Fourier transform by Ralph V. L. Hartley in 1942,

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f(x) * g(x) = \frac{F(\omega) G(\omega) + F(-\omega) G(\omega) + F(\omega) G(-\omega) - F(-\omega) G(-\omega)}{2}

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where <math>F(\omega) = \{\mathcal{H}f\}(\omega)</math> and <math>G(\omega) = \{\mathcal{H} g\}(\omega)</math>

Similar to the Fourier transform, the Hartley transform of an even/odd function is even/odd, respectively.

cas

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The properties of the Hartley kernel, for which Hartley introduced the name cas for the function (from cosine and sine) in 1942,

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  • (NB. Also translated into German and Russian.)

Further reading

  • (NB. Contains extensive bibliography.)