I am solving a very specific problem where I need to apply FFT to interpolate. I am given two sets of distinct points $A = (a_{1}, a_{2}, ... , a_{n})$ and $B = (b_{1}, b_{2}, ..., b_{n})$. Using $B$, I need to find a unique polynomial of the form $f(x) = m_{1} + m_{2}x + m_{3}x^{2} + ... n \, terms$ such that $f(a_{i}) = b_{i}; 1 \leq i \leq n$.
Can anyone provide any hints on how to do this? I know there exist FFT approaches to interpolation, but how to look for this specific interpolation that satisfies $f(a_{i}) = b_{i}; 1 \leq i \leq n$.
Will inverse DFT do it better? Or anything else that might work better here?