The Fourier Transform
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The transition from Fourier Series which were introduced in Chapter 2, to the Fourier Transform is similar to the transition from the dot product to the inner product. When dealing with functions, the question became “How many samples of the function are sufficient to approximate, accurately, the dot product that would be used to compare vectors?”. The answer is that the safe way to make sure you have enough samples is to not sample, but rather use the integral, or inner product, instead of the dot product. That way convergence issues are not an issue, and everything was certain, while preserving the geometry of the dot product.