📂Linear Algebra

Convolution's General Definition

Definition

Given the integral transform $J$ and two functions $f$, $g$, a function $f \ast g$ fulfilling the conditions below is defined as the convolution of $f$ and $g$ with respect to $J$.

$$J(f \ast g)=(Jf)(Jg)$$

Explanation

According to the definition, the convolution, being the integral transform of a product, can be divided into the product of integral transforms. This means that two functions, which were bound in a single integral, can be separated into two integrals. It’s a useful technique when finding the inverse of a forward operator expressed as an integral.

The term convolution generally refers to the convolution of the Fourier transform without specific mention.

Fourier Transform

$$(f \ast g)(x) =\int _{-\infty} ^{\infty}f(y)g(x-y)dy$$

Laplace Transform

$$(f \ast g)(x) = \int_{0}^{x}f(x-y)g(y)dy$$

Mellin Transform

$$( f\times g)(x)=\int _{0} ^{\infty} f(y)g \left( \frac{x}{y} \right)\frac{dy}{y}$$