Abstract
In low dimensions, nonlinear approximation is generally based on the existence of local regularities. The lecture shows that wavelet bases play a special role, as they allow us to obtain near-optimal approximations of functions that are locally regular. Local regularity can be specified by a Lipchitz exponent.
The wavelet transform is defined by projecting x(t) onto wavelets that are localized functions. These are deduced from a parent wavelet, which is translated and dilated, thus defining a wavelet basis. We demonstrate that the local Lipchitz regularity exponent is obtained from the local decay of the wavelet coefficients, as the scale tends towards 0.
The decomposition of a signal x(t) in a wavelet basis is also related to a sampling approximation of t. In the case where the parent wavelet is a Shannon wavelet, whose Fourier transform is the interval indicator, the decomposition into the wavelet basis is obtained from Shannon's sampling theorem.
