In the previous lesson, we have derived an ideal interpolation scheme based on the sinc function. Remember that the sinc is a bandlimited function; we will start this lesson by showing that the result of sinc interpolation is also bandlimited.

This raises the question: if any sequence can be interpolated to a bandlimited function via the sinc, is the converse also true? Namely, can any bandlimited function be converted to a discrete-time sequence The answer will be easy to prove once we recognize that the set of bandlimited functions constitutes a Hilbert space and that sampling is simply the decomposition of a function in this space into a linear combination of basis functions.

The final result will be the full statement of the sampling theorem.