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Concept# Convolution theorem

Summary

In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the pointwise product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain) equals point-wise multiplication in the other domain (e.g., frequency domain). Other versions of the convolution theorem are applicable to various Fourier-related transforms.
Functions of a continuous variable
Consider two functions g(x) and h(x) with Fourier transforms G and H:
\begin{align}
G(s) &\triangleq \mathcal{F}{g}(s) = \int_{-\infty}^{\infty}g(x) e^{-i 2 \pi s x} , dx, \quad s \in \mathbb{R}\
H(s) &\triangleq \mathcal{F}{h}(s) = \int_{-\infty}^{\infty}h(x) e^{-i 2 \pi s x} , dx, \quad s \in \mathbb{R}
\end{align}
where \mathcal{F} denotes the Fourier transform operator. The transform may be normalized in

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