We can expand any analytic around a point function in a Taylor series around the same point (I consider real functions for now).
$$f(x)=\sum_{k=0}^\infty f^{(k)}(x_0) \frac{(x-x_0)^k}{k!}$$
By using fractional calculus, can we also represent non-analytic functions in a kind of a 'Taylor integral'?
$$g(x)=\int_0^\infty g^{(a)}(x_0) \frac{(x-x_0)^a}{\Gamma(a+1)}da$$
Here $g^{(a)}$ is a fractional derivative.
My reasoning is as follows. We have Fourier series for periodic functions and Fourier integral (Fourier transform) for non-periodic functions. Could it work the same way for Taylor series?
There is a very similar question even with almost the same notation (I've just found it though). However, it doesn't use the concept of fractional derivative.