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 2019 (Monday, January 21 to Friday, March 29)
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 assessment for this course will be via a single take-home paper in April/May with 2 weeks to complete and submit online. There will be 4 questions, and you will need the equivalent of 2 questions to pass.
No assignments have been set for this course.
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