Let and define . Then with . Denote the original frequency index by , where and the new frequency index by , where .
Definition: The ideal bandlimited interpolation of a spectrum , , to an arbitrary new frequency is defined as
Note that this is just the definition of the DFT with replaced by . That is, the spectrum is interpolated by projecting onto the new sinusoid exactly as if it were a DFT sinusoid. This makes the most sense when is assumed to be samples of a time-limited signal. That is, if the signal really is zero outside of the time interval, then the inner product between it and any sinusoid will be exactly as in the equation above. Thus, for time limited signals, this kind of interpolation is ideal.
Since is initially only defined over the roots of unity, while is defined over roots of unity, we define for by ideal bandlimited interpolation.
Theorem: For any
Proof: Let with . Then