NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University
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Stretch Theorem (Repeat Theorem)
Theorem: For all,
Proof: Recall that
Let, where
,
. Also define the new denser frequency grid associated with length
by
, with
as usual. Then
But
Thus,, and by the modulo indexing of
,
copies of
goes from
to
.
