# Fractional calculus and spectral theory

I've been trying to build a more solid understanding of fractional calculus, and the closer I look, the more it seems like there are some deep connections to Fourier analysis (and spectral theory in general) that I can't quite make out.

Consider the following thought experiment:

We can consider the identity operator to be the limit of a sequence of gaussians in a convolution algebra. Similarly, we can consider the differential operator $$D$$ to be the limit of a sequence of derivatives of gaussians. These give us a pretty solid basis for visualizing what these operators look like (graphics courtesy of wikipedia):

When we extend to fractional operators, we obtain a continuous interpolation between these two:

As the order increases past $$1$$, more peaks emerge. Notice that this is beginning to look an awful lot like a wavelet basis...

Now, these operators act convolutionally - so, given some function $$f$$ in the convolution algebra, we can consider $$f'(t) = \langle D,(x \to f(x-t)) \rangle$$. In particular, we can consider $$f^{(a)}(0) = \langle D^a,f \rangle$$.

This notation seems highly suggestive - we have an indexed family of operators ($$D^a$$), and a continuous function defined by their action (via the inner product) on a given reference function. When we do this same thing with sinusoids, we get a Fourier transform. Does this yield anything similar?

Viewed another way: when we only look at derivatives of discrete order, this kind of expansion yields a power series. If fractional calculus allows us to look at derivatives of continuous order, can we generalize a power series to some sort of continuous spectrum, in the same way we generalize from a discrete Fourier series to a continuous Fourier transform?

And is there anything behind apparent similarity to wavelet transforms?

I realize this question is pretty vague, but any guidance would be appreciated.