Fourier transform of a triangular function

For ${\theta }_n(x)=(1-|x|/n{)}_{+}$,with value $n$ at zero.

The triangular functionhas 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.

The squared sinc function is nonnegative and integrable; it occurs as the Fourier transform of a triangular function.

It is the pointwise square of the sinc function, with its normalization inherited from that function.

The cutoffs ${\theta }_n(\xi )=(1-|\xi |/n{)}_{+}$ increase pointwise to one, while their Fourier transforms are nonnegative and have integral $2\pi$ under the angular-frequency convention.

If $f\in L^1\cap L^{\infty }$ and $\hat{f}\ge 0$, testing against triangular frequency cutoffs and using monotone convergence gives $\Vert \hat{f}{\Vert }_1\le 2\pi \Vert f{\Vert }_{\infty }$.