You can treat derivation as a linear operator on vector-space of functions, let's name it D. Now second and third derivatives as operators are quite simple since they are just D^2 and D^3. The question is if you can well define sqrt(D), or saying that if there is one and only one operator D_half such that D_half^2 is D (existence and unicity). It breaks at unicity since there are multiple half derivative constructions (frequency domain approach, Taylor-series approach,...).
On the other hand sqrt is well defined for positive real numbers (however not well defined for general complex numbers). There is a similar definition only for positive operators on Hilbert spaces (you pick the positive operator from all the possible square roots). However derivation is not a positive operator on the most used Hilbert-spaces.
At some point in my college Linear Algebra class I realized that the exponential function is an eigenvector of the derivative operator. Hilbert spaces are fun.
And the Gaussian distribution is an eigenvector of the Fourier transform.
Not surprisingly both of these facts correlate with how fundamental is the exponents function to solving linear differential equations, and how fundamental is the Gaussian distribution in probability theory.
Your approach seems to be simpler than the blog post. Can't you just note that the pth derivative of exp ax is a^p exp ax and then just extend via a fourier series? Maybe the non-uniqueness is too unsatisfying.
I assume you are thinking the pth derviative of exp(jkx) which is (jk)^p exp(jkx) and extending this to Fourier-transform (or series). However (j*k)^(1/2) is not so well-defined, you can pick arbitrarily from two complex values for each k. This approach leads to infinite half derivative constructions.
I remember reading a paper that claimed that integrating on a Grassmann variable space was equivalent to integrating in a space with -2 dimensions of the usual commuting variables.
When I first learned this in a graduate abstract algebra class, however, I was convinced that my professor, who was not a native English speaker, was simply mistranslating from whatever language he was used to talking about these things in. (He was, IIRC, a Romanian, educated in Germany.)
On the other hand sqrt is well defined for positive real numbers (however not well defined for general complex numbers). There is a similar definition only for positive operators on Hilbert spaces (you pick the positive operator from all the possible square roots). However derivation is not a positive operator on the most used Hilbert-spaces.