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Section: New Results

Best constrained analytic approximation

Participants : Laurent Baratchart, Sylvain Chevillard, Juliette Leblond, Dmitry Ponomarev, Elodie Pozzi.

This work is performed in collaboration with Jonathan Partington (Univ. Leeds, UK).

Continuing effort is being paid by the team to carry over the solution to bounded extremal problems of section 3.3.1 to various settings. We mentioned already in section 6.2 the extension to 2-D diffusion equations with variable conductivity for the determination of free boundaries in plasma control and the development of a generalized Hardy class theory. We also investigate the ordinary Laplacian in ${ℝ}^3$, where targeted applications are to data transmission step for source detection in electro/magneto-encephalography (EEG/MEG, see section 6.1 ).

Still, questions about the behaviour of solutions to the standard bounded extremal problems $\left(P\right)$ of section 3.3.1 deserve attention. We realized this year that Slepian functions are eigenfunctions of truncated Toeplitz operators in 2-D. This can be used to quantify robustness properties of our resolution schemes in $H^2$ and to establish error estimates, see [25] . Moreover we considered additional interpolation constraints [28] , as a simplified but already interesting issue, before getting at extremal problems for generalized analytic functions in annular non-smooth domains. The latter arise in the context of plasma shaping in tokamaks like ITER, and will be the subject of the PhD thesis of D. Ponomarev.

In another connection, weighted composition operators on Lebesgue, Sobolev, and Hardy spaces appear in changes of variables while expressing conformal equivalence of plane domains. A universality property related to the existence of invariant subspaces for these important classes of operators has been established in [19] . Additional density properties also allow one to handle some of their dynamical aspects (like cyclicity).