Exercise 4, § 3

Let $G =\mathbf{R}$. For every real number $\alpha >1$, define $f_{\alpha}\in L^1(\mathbf{R})$ by

2 if $0< x <1$

$f_{\alpha}(x) =$ 1 if $1\leqslant x < \alpha$

0 otherwise.

a) The functions $f*\varepsilon_x$ for $x\in \mathbf{R}$ form a total set in $L^1(\mathbf{R})$ if and only if $\alpha$ is irrational.

b) The functions $f*\varepsilon_x$ for $x\in \mathbf{R}$ form a total set in $L^2(\mathbf{R})$.