We study functional regression with random subgaussian design and real-valued response. The focus is on the problems in which the regression function can be well approximated by a functional linear model with the slope function being “sparse” in the sense that it can be represented as a sum of a small number of well separated “spikes”. This can be viewed as an extension of now classical sparse estimation problems to the case of infinite dictionaries. We study an estimator of the regression function...

The semi-group associated with the Cauchy problem for a scalar conservation law is known to be a contraction in $L^1$. However it is not a contraction in $L^p$ for any $p\>1$. Leger showed in [20] that for a convex flux, it is however a contraction in $L^2$ up to a suitable shift. We investigate in this paper whether such a contraction may happen for systems. The method is based on the relative entropy method. Our general analysis leads us to the new geometrical notion of Genuinely non-Temple systems. We treat in...