1 Linearity properties of the Fourier transform

(i) If $f\left(t\right)$ , $g\left(t\right)$ are functions with transforms $F\left(\omega \right),G\left(\omega \right)$ respectively, then

$\bullet \mathcal{F}\left\{f\left(t\right)+g\left(t\right)\right\}=F\left(\omega \right)+G\left(\omega \right)$

i.e. if we add 2 functions then the Fourier transform of the resulting function is simply the sum of the individual Fourier transforms.

(ii) If $k$ is any constant,

$\bullet \mathcal{F}\left\{kf\left(t\right)\right\}=kF\left(\omega \right)$

i.e. if we multiply a function by any constant then we must multiply the Fourier transform by the same constant. These properties follow from the definition of the Fourier transform and from the properties of integrals.

Examples

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2.

using the standard result for $\mathcal{F}\left\{p_a\left(t\right)\right\}$ .

Task!

If $f\left(t\right)=\left\{\begin{array}{ccc}\hfill 6\hfill & \hfill \hfill & \hfill -2\le t\le 2\hfill \\ \hfill 0\hfill & \hfill \hfill & \hfill \text{otherwise}\hfill \end{array}\right.$ write down $F\left(\omega \right).$

Answer