I haven't quite understood the relation between Fourier Transform and convolution. When should I use Fourier Transform to determine a convolution? Is it always simpler?

For example, if I want to show that $f_a*f_b=f_{a+b}$ using Fourier Transform where $f_a(x)=\frac{a}{\pi(x^2+a^2)}$. what is the simplest way of doing it?

By definition : $\mathcal{F}\{f_a*f_b\}=\mathcal{F}\{f_a\}\mathcal{F}\{f_b\}$

So do I have to use this formula? $f_a*f_b=\mathcal{F^{-1}}\{\mathcal{F}\{f_a\}\mathcal{F}\{f_b\}\}$


Using the duality property, the FT of $f_a(x)$ is of the form (plus some scaling factors) $$e^{-a|\omega|}$$ Hence, the FT of $f_a*f_b$ has the form $$e^{-a|\omega|}\cdot e^{-b|\omega|}=e^{-(a+b)|\omega|}$$ whose inverse FT has the form $\frac{a+b}{x^2+(a+b)^2}$ which is the same for $f_{a+b}$.

But again, you need to take into the consideration the scaling factors.


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