information:classification
Convolution theorem

The Fourier transform translates between convolution and multiplication of functions. If f (x) and g (x) are integrable functions with Fourier transforms f̂ (ξ) and ĝ (ξ) respectively, then the Fourier transform of the convolution is given by the product of the Fourier transforms f̂ (ξ) and ĝ (ξ) (under other conventions for the definition of the Fourier transform a constant factor may appear).

This means that if: h (x) = (f ∗ g) (x) = ∫ − ∞ ∞ f (y) g (x − y) d y, {\displaystyle h(x)=(f*g)(x)=\int _{-\infty }^{\infty }f(y)g(x-y)\,dy,} where ∗ denotes the convolution operation, then: h ^ (ξ) = f ^ (ξ) g ^ (ξ). {\displaystyle {\hat {h}}(\xi)={\hat {f}}(\xi)\,{\hat {g}}(\xi).}

In linear time invariant (LTI) system theory, it is common to interpret g (x) as the impulse response of an LTI system with input f (x) and output h (x), since substituting the unit impulse for f (x) yields h (x) = g (x). In this case, ĝ (ξ) represents the frequency response of the system.

Conversely, if f (x) can be decomposed as the product of two square integrable functions p (x) and q (x), then the Fourier transform of f (x) is given by the convolution of the respective Fourier transforms p̂ (ξ) and q̂ (ξ).