Abstract
The lecture begins by showing the difference between linear and nonlinear approximations in orthonormal bases. Linear approximation of a signal x is achieved by selecting a limited number M of decomposition coefficients in a basis, whereas nonlinear approximation can adapt the choice of these M coefficients as a function of x, in particular by choosing the largest ones. An example of a linear approximation is the uniform sampling of a function, as opposed to adaptive, non-linear sampling, which adapts to the function's local regularity.
In a linear framework, the low-dimensional approximation problem is tackled by studying the underlying regularity. The regularity of a function x(t) can be defined by the existence of k derivatives of finite energies, which corresponds to Sobolev regularity. The properties of the derivatives are characterized in a basis that diagonalizes the derivation operator. As this operator is covariant by translation, we demonstrate that it is the Fourier basis. The lecture briefly reviews the properties of bases and the Fourier integral. Sobolev regularity is defined on the one-dimensional Fourier integral.
