Convolution power

If $x$ is a function on Euclidean space and $n$ is a natural number, then one can define the convolution power as follows:

$x^{*n}\triangleq \underbrace {x*x*x*\cdots *x*x} _{n}$

where * denotes the convolution operation. This definition makes sense if x is an integrable function (in L¹), or is a compactly supported distribution.

Derivatives

From the properties of convolution, the following holds:

${\mathcal {D}}{\big \{}x^{*n}{\big \}}=x*{\mathcal {D}}{\big \{}x^{*(n-1)}{\big \}}$

where ${\mathcal {D}}$ denotes the derivative operator.

See also

• Convolution
• Convolution theorem
• Fourier transform
• Taylor series