The convolution theorem establishes that discrete convolution of sequences corresponds to pointwise multiplication in the frequency domain via the Discrete Fourier Transform (DFT), reducing computational complexity from $O(n^2)$ to $O(n \log n)$. This operation combines two functions or probability distributions by reversing one sequence, shifting it across the other with overlapping offsets, and summing pairwise products. The concept is fundamental in signal processing and probabilistic theory as a mechanism for merging independent random variables or performing linear filtering operations efficiently through polynomial multiplication properties.
The convolution theorem establishes that discrete convolution of sequences corresponds to pointwise multiplication in the frequency domain via the Discrete Fourier Transform (DFT), reducing computati…