Example

We take $G = \mathbf{R}$. Let us define a function $F_n \in \mathscr{K}(\mathbf{R})$ by

Let $A_n = \int_{-1}^{+1} F_n(x) dx$, and $G_n = A_n^{-1} F_n$. It is immediate that the measures $G_n(x) dx$ satisfy the conditions of §2, No. 7, Cor. 1 of Lemma 4. Let $\mu$ be a measure on $\mathbf{R}$ whose support is contained in $[-1/2, 1/2]$. Then If $-1/2 \leq x \leq 1/2$, then therefore $\mu * G_n$ coincides on $[-1/2, 1/2]$ with a polynomial. In particular, if $f$ is a continuous function with support contained in $[-1/2, 1/2]$, then $f * G_n$ coincides in $[-1/2, 1/2]$ with a polynomial; moreover, by Prop. 5 (iv), and §2, No. 7, Cor. 3 of Lemma 4, $f * G_n$ converges uniformly to $f$. If $f$ is of class $C^r$, the derivatives $D^s(f * G_n)$ tend uniformly to $D^s f$ for $0 \leq s \leq r$.