Interference of traveling waves can be used to model the action of fractional integro-derivatives.

The FIDs can be regarded as an interpolation of the integral derivatives of the function acted upon--to be more precise, a sinc function interpolation, as in the Whittaker-Shannon interpolation formula, of the integral derivatives, properly normalized .

The FID can be expressed as

$$$$
$\displaystyle\frac{d^{\beta}}{dx^\beta}f(x)=\lim_{\:r \rightarrow 1^{-}}\frac{1}{2\pi i}\displaystyle\oint_{|z-x|=r \cdot x}f(z)\frac{\beta!}{(z-x)^{\beta+1}}dz=FP\displaystyle\int_{0}^{x}f(z)\frac{\;\;\;\;\;(x-z)^{-\beta-1}}{(-\beta-1)!} dz$,
$$$$

where $x>0$, $(z-x)=r \cdot x e^{i\theta}$ in the contour integral, $-\pi<\theta<\pi$, $FP$ denotes a Hadamard-type finite part, and $\beta$ is real,

or, with a change of variable and notation and using $\binom{0}{\beta}=\frac{1}{\beta!(-\beta)!}=\frac{\sin (\pi \beta)}{\pi \beta}$, formally

$$$$
$\displaystyle D^{\beta} f(x)=FP\displaystyle\int_{0}^{x}f(x-z)\frac{\;\;\;\;\;z^{-\beta-1}}{(-\beta-1)!} dz=FP\displaystyle\int_{0}^{x}\frac{\;\;\;\;\;z^{-\beta-1}}{(-\beta-1)!}exp(-zD)\; dz\; f(x)$
$$$$

$=\displaystyle\sum_{n=0}^{\infty } (-1)^n
\frac{f^{(n)}(x)}{n!}\frac{1}{(-\beta-1)!}\frac{x^{n-\beta}}{n-\beta}=\frac{\beta!}{x^{\beta}}\sum_{n=0}^{\infty }\frac{x^{n}}{n!}f^{(n)}(x)\frac{sin(\pi(n-\beta))}{\pi(n-\beta)}.$
$$$$

Summarizing, we have an interpretation of a FID as the sinc fct. interpolation

$$$$
$$\displaystyle \frac{x^{\beta}}{\beta!}D^{\beta} f(x)=\sum_{n=0}^{\infty }\frac{x^{n}}{n!}f^{(n)}(x)\frac{sin(\pi(n-\beta))}{\pi(n-\beta)}.$$
$$$$

For $f(x)=x$ and $\beta=\frac{1}{2}$, this gives
$$$$
$$\displaystyle \frac{x^{\frac{1}{2}}}{(\frac{1}{2})!}D^{\frac{1}{2}} x=x\;[\frac{sin(\pi(-\frac{1}{2}))}{\pi(-\frac{1}{2})}+\frac{sin(\pi(\frac{1}{2}))}{\pi(\frac{1}{2})}]=x\;\frac{4}{\pi},$$

or

$$\displaystyle D^{\frac{1}{2}} x= {\frac{1}{2}}!\;\frac{4}{\pi}x^{\frac{1}{2}}=\;\frac{2}{\sqrt{\pi}}x^{\frac{1}{2}}=\frac{x^{\frac{1}{2}}}{(\frac{1}{2})!}.$$
$$$$

Construct traveling pulses from plane waves over an inverse wavelength ($k=1/\lambda$) bandwidth of $k_0$:
$$$$
$$\displaystyle \frac{x}{k_0}\int_{-\frac{k_0}{2}}^{\frac{k_0}{2}}cos(2\pi k(u-vt+n/k_0)) dk=x\frac{sin(\pi k_0(u-vt+n/k_0))}{\pi k_0(u-vt+n/k_0)}$$
$$$$

where $n=0$ or $1$, and $u, v, t,$ and $x$ are the spatial coordinate, wave velocity, time, and wave amplitude, respectively.

The interference of the the pulses at $u=0$ for $t=\frac{\beta}{k_0 v}$ is

$$$$
$$\displaystyle x\;[\frac{sin(\pi(-\beta))}{\pi(-\beta)}+\frac{sin(\pi(1-\beta))}{\pi(1-\beta)}]=\frac{x^\beta}{\beta!}D^{\beta} x.$$
$$$$

For $f(x)=x^m$, $m+1$ traveling sinc pulses are required with the amplitude of the $n$'th pulse being $\binom{m}{n}x^m$.

For $f(x)=x^\alpha$, the interpolation formula gives, for $Re(\alpha)>-1$,

$$$$
$$\displaystyle \frac{x^{\beta}}{\beta!}D^{\beta} x^\alpha=x^\alpha\displaystyle\sum_{n=0}^{\infty }\binom{\alpha }{n}\frac{\sin (\pi (\beta -n))}{\pi (\beta -n)}=\binom{\alpha }{\beta}x^\alpha,$$

or

$$\displaystyle D^{\beta} \frac{x^\alpha}{\alpha!}=\frac{x^{\alpha-\beta}}{(\alpha-\beta)!}.$$
$$$$

Note:

1) $x^\alpha$ is an eigenfunction of the operator $x^\beta D^\beta$, so we can expect the inverse Mellin transform to aid in characterizing the operator's action on various functions. In fact, this interpolation perspective is consistent with the discussion in MO79868 (What does Mellin inversion really mean?) of the Mellin transform as providing an interpolation of the coefficients of the Taylor series of a function (in this case, the function $w(z)=H(x-z)f(x-z)$ where $H(x)$ is the Heaviside step function), a perspective on Ramanujan's Master Formula / Theorem.

2) All of this, of course, is related to the theory of Green's functions and impulse response functions in signal processing. And, the final example concerns essentially the beta function, which is related to scattering amplitudes in string theory.)

3) It seems a little misleading to distinguish between "locality" and "non-locality" based on the singularities of the gamma fct. From the contour integral rep, the distinction is maybe better understood as the difference between pole singularities and branch cuts for the "Cauchy" kernel.