A fundamental tool in practical spectrum analysis is zero
padding. This theorem shows that zero padding in the time domain
corresponds to ideal interpolation in the frequency domain (for
truly time-limited signals):

Theorem: For any

where
was defined in Eq. (7.4), followed by the
definition of
.

Proof: Let with . Then

Thus, this theorem follows directly from the definition of the ideal
interpolation operator
. See §8.1.3 for an
example of zero-padding in spectrum analysis.