3. [M25] Consider the following minimization problem10 E(f ) =

+1

−1 f 2(x)

1 − f

(x)

2 dx (4.4.107)

where the function f : [−1. .1] → R is expected to satisfy the boundary conditions f (−1) = 0 and f (0) = 1. Prove that there is no solution in C1, whereas, when allowing one discontinuity of f

, the function f (x) = x · [x ≥ 0] minimizes E.

#!# 4. [M26] Let us consider the case of functions f : R → R and the differential operator L =

∞

κ=0

(−1)κ Σ2κ

κ!2κ

· d2κ

dx2κ

, which is associated with the Gaussian kernel. Prove that limx→0 Lg =∞.