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Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



Wavelet transforms versus Fourier transforms

Author: Gilbert Strang
Journal: Bull. Amer. Math. Soc. 28 (1993), 288-305
MSC: Primary 42C15; Secondary 65T20, 94A11
MathSciNet review: 1191480
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Abstract: This note is a very basic introduction to wavelets. It starts with an orthogonal basis of piecewise constant functions, constructed by dilation and translation. The "wavelet transform" maps each f(x) to its coefficients with respect to this basis. The mathematics is simple and the transform is fast (faster than the Fast Fourier Transform, which we briefly explain), but approximation by piecewise constants is poor. To improve this first wavelet, we are led to dilation equations and their unusual solutions. Higher-order wavelets are constructed, and it is surprisingly quick to compute with them -- always indirectly and recursively.

We comment informally on the contest between these transforms in signal processing, especially for video and image compression (including high-definition television). So far the Fourier Transform -- or its 8 by 8 windowed version, the Discrete Cosine Transform -- is often chosen. But wavelets are already competitive, and they are ahead for fingerprints. We present a sample of this developing theory.

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Keywords: Wavelets, Fourier transform, dilation, orthogonal basis
Article copyright: © Copyright 1993 American Mathematical Society