Convolution power

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If $\ x$ is a function 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.

More generally, we can define convolution power for any complex number $\ c$ :

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

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.

1 Convolution root
2 Convolution exponential and logarithm
3 Other definitions
4 Convolution inverse
5 Derivatives
6 See also

[edit] Convolution root

We define the convolution root as

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

This obeys the following property:

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

[edit] Convolution exponential and logarithm

We define the convolution exponential and convolution logarithm as follows:

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

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

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

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:

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

[edit] Other definitions

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

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

[edit] Convolution inverse

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

$\ x^{*-1} = \sum_{k=0}^\infin (x-\delta)^{*k} \,$

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

Proof:

$\begin{align} x * \sum_{k=0}^\infin (x-\delta)^{*k} &= \mathcal{F}^{-1} \left\{ \mathcal{F} \{x\} \, \mathcal{F} \left\{ \sum_{k=0}^\infin (x-\delta)^{*k} \right\} \right\} \\ &= \mathcal{F}^{-1} \left\{\mathcal{F}\{x\} \, \sum_{k=0}^\infin \big(\mathcal{F}\{x\}-\mathcal{F}\{\delta\}\big)^{k} \right\} \\ &= \mathcal{F}^{-1} \left\{\mathcal{F}\{x\} \, \sum_{k=0}^\infin \big(\mathcal{F}\{x\}-1\big)^{k} \right\} \\ &= \mathcal{F}^{-1} \left\{\frac{\mathcal{F}\{x\} }{ \mathcal{F}\{x\}} \right\} \\ &= \delta \end{align}$