I was wondering what the physical insight is of integrating a product of two functions. When we do that for a Fourier transform, we decompose a function into its constituent frequencies, and that's because the exponential with an imaginary x in the transform can be seen as a weighting function that looks at the frequency in the original signal. I'm not sure if this kind of logic is accurate. In particular, what caused this question is the consideration of the Riemann-Liouville fractional derivative (definitions can be seen here: http://www.hindawi.com/journals/mpe/2014/238459/). We are integrating a function with respect to a power law (x - ε)n - α - 1 so it's a similar case. Does the power law "weigh" the information of the function it multiplies? Is there another way to think about it? Any help is appreciated!