They first describe the attendant conformism that makes most of the engineering world live in L2 world (FEM).
and then provide what will sell the idea
Sweet. As they say in the concluding remarks: "We should not be afraid to work in Banach spaces. It is doable." It mirrors what I say to others when mentioning compressed sensing, "Come work on the dark side of the Force":-). Further reference on these results can be found in the following two papers:
- J.-.L. GUERMOND, A finite element technique for solving first order PDE's in Lp, SIAM J. Numer. Anal. 42 2 (2004) 714--737.
- J.-.L. GUERMOND, B. POPOV, Linear advection with ill-posed boundary conditions via L1-minimization, International Journal of Numerical Analysis and Modeling, 4:1 (2007) 39-47.
On a related note, Jean-Luc Guermond Bojan Popov and Ron DeVore organize a workshop on Nonlinear Approximation Techniques Using L1 at Texas A&M University on May 16th to 18th, 2008. It'll take place in the Blocker building, Room 120.
L1-based techniques are emerging in different fields of applied sciences. The objective of the workshop is to gather leading experts that are possibly unaware of each other's activity so that they exchange ideas and get a more global view of this new set of methods. The purpose of this workshop is also stimulate cross fertilization.