Archive | May 2016

# Continuous derivative of the Fourier transform of a function

Define to be the set of functions which are continuous, bounded, and satisfy

Then given , define its Fourier transform to be

In my readings of the Fourier transform I encountered details about its derivative. In particular, if , then is continuously differentiable and . The proof I read of it left much to be desired, or at least much was left to the reader. I ended up putting considerable effort into a complete proof, using various resources from internet searches, and I thought I would give back with this proof itself. Link to proof: Continuous derivative of the Fourier transform of a function