**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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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## The Fourier Theorems

In this section the main Fourier theorems are stated and proved. It is no small matter how simple these theorems are in the DFT case relative to the other three cases (DTFT, Fourier transform, and Fourier series). When infinite summations or integrals are involved, the conditions for the existence of the Fourier transform can be quite difficult to characterize mathematically. Mathematicians have expended a considerable effort on such questions. By focusing primarily on the DFT case, we are able to study the essential concepts conveyed by the Fourier theorems without getting involved with mathematical difficulties.

Subsections

- Linearity
- Conjugation and Reversal
- Symmetry
- Shift Theorem
- Convolution Theorem
- Dual of the Convolution Theorem
- Correlation Theorem
- Power Theorem
- Rayleigh Energy Theorem (Parseval’s Theorem)
- Stretch Theorem (Repeat Theorem)
- Decimation Theorem (Aliasing Theorem)
- Zero Padding Theorem
- Bandlimited Interpolation in Time