Classical Wavelet Theory (MAGIC094)
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Developed mostly in the 1980s, wavelets provide an alternative to Fourier series with better localization properties, and have found applications in approximation, signal and image processing, areas of applied mathematics such as acoustics and electromagnetism, and also in statistics. This course gives a non-technical introduction to wavelets, focusing on the simplest examples, such as the Haar wavelets (which go back to 1909) and the Littlewood-Paley wavelets (based on ideas from the 1930s). It will also discuss windowed Fourier transforms and wavelet transforms, as ways of capturing local behaviour of functions/data.
Spring 2017 (Monday, January 23 to Friday, March 31)
Some experience of Fourier series, Fourier transforms, and Hilbert spaces.
1. Introduction and revision of Fourier series and transforms. (1) 2. The Haar wavelet and the idea of a multiresolution expansion. (2) 3. Paley-Wiener spaces, the sampling theorem, and Littlewood-Paley wavelets. (2) 4. Riesz bases and frames. (2) 5. Windowed Fourier transforms, Heisenberg's inequality, and wavelet transforms. (3)
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The course will be assessed by a take-home examination in April. There will be four questions, and you will need to obtain 50 percent to pass.
No assignments have been set for this course.
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