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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, as defined in Appendix B). 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.

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[How to cite this work] [Order a printed hardcopy]

``Mathematics of the Discrete Fourier Transform (DFT), with Music and Audio Applications'', by Julius O. Smith III, W3K Publishing, 2003, ISBN 0-9745607-0-7.
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA  [Automatic-links disclaimer]