I have from and to read about different definitions of fractional derivatives. That is functionals $${\bf D}^{\frac 1 k}$$ that take (often an integer) $k$ so that: $${\left({\bf D}^{\frac 1 k}\right)}^k = {\bf D}$$
So for example doing a one-third derivative three times should give us the same thing as an ordinary differential operator would ( for all functions in some set of differentiable functions ).
Consider a basis of $c_1 \sin(nx) + c_2 \cos(nx)$ with coefficients in a:
$$Dv = \left(\begin{array}{cc}0&-n\\n&0\end{array}\right)\left(\begin{array}{c}c_1\\c_2\end{array}\right) = n\left(\begin{array}{cc}0&-1\\1&0\end{array}\right)\left(\begin{array}{c}c_1\\c_2\end{array}\right)$$
We can now define as a solution to the equation $({D^\frac 1 k})^{k} = D$
What criteria can we make to make such an operation sensible? We could fill out with arbitrary functions which still fulfill the equation and just stuff them in between. For example a double size space:
$$D^{1/3}v = \sqrt[3]{n}\left(\begin{array}{cc}0&0&-1&0\\0&0&0&-i\\1&0&0&0\\0&i&0&0\end{array}\right)\left(\begin{array}{c}0\\c_1\\0\\c_2\end{array}\right) \Leftrightarrow \left(D^{1/3}\right)^3v = n\left[\begin{array}{cccc}0&0&-i&0\\0&0&0&-1\\i&0&0&0\\0&1&0&0\end{array}\right]$$
We see that for exponent 3 column 2 and 4 map on each other with $-n,n$ respectively like they should if they were the basis functions from before. But other than that, what stops us from filling out with arbitrary functions? We know they must be $in$ and $-in$ in terms of each other when doing derivative, is that enough of a constraint to enforce them to be any particular functions or do we have freedom in choosing them?