Classical Fourier analysis and the modern theory of wavelets are widely used today in such applications as signals analysis and image processing. While the classical theory remains ideal for the analysis and processing of periodic phenomena, the use of wavelets is much more efficient in connection with irregular and fractal structures. This module provides a careful balance between the underlying theory and the practical algorithms which are used in applications.
Review of Fourier series. Fourier transforms, inversion, convolution product, Parseval's identity, Fourier transforms of Gaussians. Physical interpretation: spectral content of a signal.
Physical motivation: deficiencies of Fourier transforms, Wavelet functions, translations, dilations, Wavelet series and transforms, orthonormal families, Window functions and their parameters, Time-frequency analysis, the Uncertainty Principle, Inverse formulae and duals, Examples of spline-wavelets.
Lectures supported by tutorials
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