### Zero Padding Theorem (Spectral Interpolation)

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 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.
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