Editing Hartley transform (section)

==Definition==
The Hartley transform of a [[function (mathematics)|function]] <math>f(t)</math> is defined by:

<math display=block>
H(\omega) = \left\{\mathcal{H}f\right\}(\omega) =  \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty
f(t) \operatorname{cas}(\omega t) \, \mathrm{d}t\,,
</math>

where <math>\omega</math> can in applications be an [[angular frequency]] and

<math display=block>
\operatorname{cas}(t) = \cos(t) + \sin(t) = \sqrt{2} \sin (t+\pi /4) = \sqrt{2} \cos (t-\pi /4)\,,
</math>

is the cosine-and-sine (cas) or ''Hartley'' kernel. In engineering terms, this transform takes a signal (function) from the time-domain to the Hartley spectral domain (frequency domain).

=== Inverse transform ===
The Hartley transform has the convenient property of being its own inverse (an [[Involution (mathematics)|involution]]):

<math display=block>f = \{\mathcal{H} \{\mathcal{H}f \}\}\,.</math>

=== Conventions ===

The above is in accord with Hartley's original definition, but (as with the Fourier transform) various minor details are matters of convention and can be changed without altering the essential properties:
*Instead of using the same transform for forward and inverse, one can remove the <math>{1}/{\sqrt{2\pi}}</math> from the forward transform and use <math>{1}/{2\pi}</math> for the inverse&mdash;or, indeed, any pair of normalizations whose product is {{nowrap|1=<math>{1}/{2\pi}</math>.}} (Such asymmetrical normalizations are sometimes found in both purely mathematical and engineering contexts.)
*One can also use <math>2\pi\nu t</math> instead of <math>\omega t</math> (i.e., frequency instead of angular frequency), in which case the <math>{1}/{\sqrt{2\pi}}</math> coefficient is omitted entirely.
*One can use <math>\cos-\sin</math> instead of <math>\cos+\sin</math> as the kernel.