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.

More generally, the convolution power of x can be defined for any complex number $c$ by

$x^{*c}\triangleq {\mathcal {F}}^{-1}{\Big \{}{\mathcal {F}}\{x\}^{c}{\Big \}}$

where ${\mathcal {F}}$ denotes the Fourier transform, and ${\mathcal {F}}^{-1}$ the inverse Fourier transform. The convolution theorem shows that the previous definition for natural numbers holds.

Convolution root

We define the convolution root as

${\sqrt[{*c}]{x}}\triangleq x^{*(1/c)}$

This obeys the following property:

$\left({\sqrt[{*c}]{x}}\right)^{*c}=x$

Convolution exponential and logarithm

We define the convolution exponential and convolution logarithm as follows:

${\begin{aligned}\exp ^{*}(x)&\triangleq {\mathcal {F}}^{-1}{\Big \{}\exp {\big (}{\mathcal {F}}\{x\}{\big )}{\Big \}}\\\ln ^{*}(x)&\triangleq {\mathcal {F}}^{-1}{\Big \{}\ln {\big (}{\mathcal {F}}\{x\}{\big )}{\Big \}}\end{aligned}}$

Using the convolution theorem and the Taylor series expansions for $\exp(x)$ and $\ln(x)$, we find that the convolution exponential and logarithm may also be expressed as

${\begin{aligned}\exp ^{*}(x)&=\sum _{k=0}^{\infty }{\frac {x^{*k}}{k!}}\\\ln ^{*}(x)&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n+1}}(x-\delta )^{*(n+1)}\end{aligned}}$

where $\delta$ is the Dirac delta.

The convolution exponential and logarithm obey many of the same properties of the standard exponential and logarithm, but with multiplication replaced by convolution:

${\begin{aligned}\exp ^{*}(x+y)&=\exp ^{*}(x)*\exp ^{*}(y)\\\ln ^{*}(x*y)&=\ln ^{*}(x)+\ln ^{*}(y)\\\exp ^{*}(\ln ^{*}(x))&=x\end{aligned}}$

Other definitions

We may also define the following, where $x$ and $y$ are both functions:

${\begin{aligned}\ x^{*y}&\triangleq e^{\log ^{*}(x)*y}\\{\sqrt[{*x}]{y}}&\triangleq e^{\log ^{*}(y)*x^{*-1}}\\\log _{x}^{*}(y)&\triangleq {\frac {\ln ^{*}(y)}{\ln ^{*}(x)}}\end{aligned}}$

Convolution inverse

Again using the convolution theorem and the Taylor series expansion for $(1+x)^{-1}$, we find the following relationship:

$x^{*-1}=\sum _{k=0}^{\infty }(x-\delta )^{*k}\,$

This is the convolution inverse, such that $x*\sum _{k=0}^{\infty }(x-\delta )^{*k}=\delta$.

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