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

If x is itself suitably differentiable, then the properties of convolution, one has

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

where ${\mathcal {D}}$ denotes the derivative operator. Specifically, this holds if x is a compactly supported distribution or lies in the Sobolev space W^1,1 to ensure that the derivative is sufficiently regular for the convolution to be well-defined.

See also

• Convolution
• Convolution theorem
• Fourier transform
• Taylor series