I want to prove that any $L^p$ function can be approximated with a sequence converging weakly but not strongly, for this: Let $f \in L^p((0, 1))$, with $p \in [1, \infty)$. Construct a sequence of continuous functions $(f_n)_{n∈N} \subset L^p((0, 1)) ∩ C^0((0, 1))$ such that $f_n \rightarrow f$ weakly in $L^p((0, 1))$ but not strongly.
Hints:
- use that any $L^p$ function can be approximated in the strong norm with a sequence of continuous functions
- use that a periodic perturbation converges weakly but not strongly to its average ( Riemann-Lebesgue Lemma)
The Riemann-Lebesgue Lemma: Let $p \in [1, \infty], \Omega \subset \mathbb{R}^N$ be an open bounded set, and $Q \subset \mathbb{R}^N$ be an open cube. Let $f \in L^p(Q; \mathbb{R}^M)$. Extend $f$ to the whole $\mathbb{R}^N$ in a Q-periodic way: $f(z) := f (y)$, with $z \in$ $\mathbb{R}^M$ written as $z = y + q$ for some $y \in Q$ and $q \in$ $\mathbb{Z}^N$. For $n \in \mathbb{N}$ define $f_n \in$ $L^p(\Omega;\mathbb{R}^M)$ as $f_n(x) := f(nx)$.
Then,
(i) $f_n$ converges to $\bar f$ weakly in $ L^p(\Omega;\mathbb{R}^M)$, if $p \in [1, ∞)$;
(ii) $f_n$ converges to $\bar f$ weakly∗ in $L^\infty(\Omega;\mathbb{R}^M)$, if $ p = ∞$
with $\bar f$ the average of $f$ in $Q$
My ideas:
I guess the first hint refers to the use of mollifiers, so I can define $f_\varepsilon=f*\varphi_\varepsilon,$ in $\Omega_\varepsilon$ ( which should be well defined since $f \in L^p(\Omega)\subset L^1_{\text{loc}}(\Omega)$ ) where:
$\varphi_\varepsilon(x)=\frac{1}{\varepsilon}\varphi(\frac{x}{\varepsilon}), x \in \mathbb{R}$
$\varphi$the standard mollifier: $\varphi(x)=ce^{\frac{1}{x^2-1}},$ if $|x|<1; 0 $ otherwise (But I guess I have to change it a bit, because my domain is not centered at $0$, but am not very sure how. What about if I put $2x-1$ instead of $x$? like this: $\varphi(x)=ce^{\frac{1}{(2x-1)^2-1}}, x\in(0,1); 0$ otherwise
$\Omega_\varepsilon=\{x\in\Omega: \text{dist} (x,\partial \Omega)> \epsilon \}=(\varepsilon,1- \varepsilon)$
$\varphi_\varepsilon \in C_c^\infty(\mathbb{R}) \subset C^0((\mathbb{R}))$ and I think it would hold if I take $\Omega=(0,1) $ instead of $\mathbb{R}$?
Taking $\varepsilon =1/n, $ $f_n(x):=f(nx)=\varphi_{1/n}(x)=n\varphi(nx), x \in \mathbb{R}$
I am not sure to which function I should apply the lemma to get the weak convergence in $L^p((0,1))$ to f, considering that it yields convergence to an average so for sure I can't apply it to f . Besides the lemma says nothing about the non strong convergence. Could you clear up these ideas? How do I proceed from here?