Let's say I have an operator $\hat{A}$ that acts as a Fourier multiplier with some function f(k) such that application of this operator on a function g(k) in Fourier space is f(k)*g(k). An example would be the second derivative where f(k) would be $-k^2$ and application to g(k) becomes $-k^{2}*g(k)$.
My question is that if I have this operator are there any conditions on the nature of the Fourier multiplier f(k) that make it impossible for me to apply it numerically using the following method.
I take a function I want to apply my operator to $\psi(x)$ and Fourier transform it to bring it to Fourier space. This turns it into $\phi(k)$.
I then apply my Fourier multiplier.
I then apply the inverse Fourier transform to bring it back to my initial position space.
My first thoughts were that a Fourier multiplier that is discontinuous or unbounded would cause great problems. But I am honestly unsure of how to even start trying to prove if a certain Fourier multiplier would not cause any problems. I was thinking that using the Nyquist theorem would be important in formulating a solution to this but I am unsure. The Fourier multiplier I am most interested in is $|k|^\alpha$ where $\alpha \in (1,2)$. Thank you for reading.