= Triangular function
{title2=$\operatorname{tri}$}
{wiki}
= Triangular pulse
{synonym}
The triangular function
$$
\operatorname{tri}(x)=(1-|x|)\mathbf1_{[-1,1]}(x)
$$
has angular-frequency Fourier transform $4\sin^2(\xi/2)/\xi^2$.
Equivalently, this is the <convolution> of two unit-width <rectangular functions>: the overlap length is $1-|x|$ when $|x|<1$, and zero otherwise. Time rescaling and amplitude changes give <triangular pulses> with different widths and heights. The <convolution theorem> also obtains the displayed transform by squaring $\sin(\xi/2)/(\xi/2)$, with continuous value one at zero.
Back to article page
