The course is conducted in English。

This course provides graduate students with a panorama of functional analysis and approximation theory in multiple dimensions, adopting a systematic dual point of view (functions defined through a collection of measurements, weak formulations). The emphasis will be laid on the simplest, albeit modern mathematical concepts and mechanisms, with a view to avoid extraneous formalism and more abstract (e.g., topological) considerations. This knowledge will be used to model engineering problems (e.g., data acquisition, sampling), to devise methods for solving exactly or approximately the inverse problems that are related (e.g., resulting from partial differential equations), and to analyze the error resulting from the approximations.

Equip postgraduate students with advanced knowledge on functional analysis (in particular, on measure theory, integration and generalized functions), and on the approximation of functions using bases (in particular, polynomial, spline and wavelet approximations). At the end of this course, students are expected to be know how to deal with the measurements (generalized samples) of functions to construct accurate approximating functions.

Grading: 5 Homeworks on • Lebesgue integration (15%) • Hilbertian Analysis (15%) • Distribution Theory (15%) • Calculus of Variations (15%) • Approximation Theory (15%) 1 Matlab assignment on approximation theory and wavelets (20%) Class participation (5%)

D.M.A. Bressoud, A radical approach to Lebesgue’s theory of integration, Cambridge University Press, (2008) C.F. Gerald and P.O. Wheatley, Applied Numerical analysis, Addison-Wesley (1999) Gel'fand, Izrail Moiseevich, et al. Generalized functions. Vol. 1. New York: Academic press, 1968. D.C. Champeney, A handbook of Fourier theorems, Cambridge University Press (1987) R. Dautray and J.L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology: Functional and Variational Methods Springer (2000) Mallat, Stéphane. A wavelet tour of signal processing. Access Online via Elsevier, 1999.