Fourier Decomposition is a form of decomposing a [[Function]] $f$ on an interval $[a,b]$ into a (often infinite) [[Summation]] of [[Sinusoidal Functions]]. This can be seen as a [[Change of Basis Matrix|change of basis]] to represent $f$ as a [[Linear Combination]] of [[Fourier Basis|Fourier basis vectors]]. $\huge \mathcal{F}\left\{ {f} \right\}( \omega ) \triangleq \int_{-\infty}^{\infty} e^{-i \omega t} f(t)\mathrm{d}t \\ $ The Fourier [[Function|Transform]] is a [[Linear Transformation|Linear]] [[Operation|Operator]] transforming a function of [[Time Domain]] (or spatial, etc) to one of [[Frequency]] [[Domain]]. The value of the Fourier Transform at some [[Real Numbers|Real Number]] $\omega$ is equal to the strength of a certain frequency component ($f= \frac{\omega}{2\pi}$) present in the original signal $f$. $\huge \begin{align} \mathcal{F}^{-1}\left\{ f \right\} (t) \triangleq \frac{1}{2\pi} \int_{-\infty}^{\infty} \mathcal{F}\left\{ f \right\}(t) e^{i\omega t}(\omega) \mathrm{d} \omega \end{align} $