**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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 are generated as goes from to .