That is a very 'signals' based interpretation of the convolution theorem [1] applied to binary functions. In essence, the fourier-transform is based on convolu

That is a very 'signals' based interpretation of the convolution theorem [1] applied to binary functions. In essence, the fourier-transform is based on convolutions. F(eta) is essentially the convolution of f(x) with sin(eta x).* This is very loosely why the convolution theorem works. [1] https://en.wikipedia.org/wiki/Convolution_theorem https://en.wikipedia.org/wiki/Convolution_theorem * This excludes all cosine parts of the transform. Its neater to work in the complex domain and state that: F(eta) = f(x) convolved with e^(eta i x)