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Wavelets: Theory and Applications by A. Rieder, D. Maass, A. K. Louis

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Appendix

The Fourier Transform

The Fourier transform appears as a fundamental tool in this book, so its main results and properties have been summarized here. Detailed studies of Fourier analysis can be found in the books of Rudin [Rud79] and Yosida [Yos64].

In the following let f be a fast-decaying function in image or a function in image. The Fourier transform of f is given by

uneq1

where images. The scalar product in images is denoted by 〈 f, gL 2 = images. We have Parseval’s relation

uneq2

Thus the Fourier transform can be extended to images and even to the space of tempered distributions images. By Parseval's relation, the Fourier transform ...

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