Properties of Fourier Transform/Linearity

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Theorem

Let $\map f x$ and $\map g x$ be Lebesgue integrable functions.


Let $a$ and $b$ be complex numbers.

Let $\map h x$ be a Lebesgue integrable function such that:

$\map h x = a \map f x + b \map g x$


Then:

$\map {\hat h} s = a \map {\hat f} s + b \map {\hat g} s$

where $\map {\hat h} s$, $\map {\hat f} s$ and $\map {\hat g} s$ are the Fourier transforms of $\map h x$, $\map f x$ and $\map g x$ respectively.


Proof

\(\ds \map {\hat h} \zeta\) \(=\) \(\ds \int_{-\infty}^\infty e^{-2 \pi i x s} \map h x \rd x\)
\(\ds \) \(=\) \(\ds \int_{-\infty}^\infty e^{-2 \pi i x s} \paren {a \map f x + b \map g x} \rd x\)
\(\ds \) \(=\) \(\ds a \int_{-\infty}^\infty e^{-2 \pi i x s} \map f x \rd x + b \int_{-\infty}^\infty e^{-2 \pi i x s} \map g x \rd x\) Linear Combination of Definite Integrals
\(\ds \) \(=\) \(\ds a \map {\hat f} s + b \map {\hat g} s\)

$\blacksquare$


Sources

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