Discrete Fourier Transform

The document explains how to multiply polynomials efficiently using the Fast Fourier Transform (FFT). Traditional polynomial multiplication has a time complexity of (O(N^2)). FFT offers an (O(N \log N)) solution by converting polynomials from their coefficient representation to a value representation. This involves sampling the polynomials at specific points, performing pairwise multiplication of these samples, and then using interpolation to reconstruct the resulting polynomial. The core of the FFT algorithm relies on evaluating polynomials at the roots of unity. The inverse process, interpolation, is achieved using the Inverse Discrete Fourier Transform (IDFT), which is closely related to the DFT. The document also provides a recursive implementation of FFT and IFFT, demonstrating how a single function can handle both with a small modification. The method is tested on an example, showing accurate polynomial multiplication.