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Prob. Let $f\in C^1(\mathbb{R})\cap L^2(\mathbb R)$ such that $f'\in L^2(\mathbb{R})$. Then approximate $f,f'$ by $f_n,f_n'$, where $f_n\in C_c^1(\mathbb{R}).$

Possible Hope: We know that $C_c^1(\mathbb{R})$ (compactly supported one time continuously differentiable function) is dense in $L^2(\mathbb{R})$. So there exist $f_n\in C_c^1$ such that $\|f_n-f\|_2\to 0$ as $n\to \infty$. Since we assume that $f'\in L^2$, so my guess is that we can give the existence of $f_n\in C_c^1(\mathbb R)$ such that

  1. $\|f_n-f\|_2\to 0$ and
  2. $\|f_n'-f'\|_2\to 0$ as $n\to \infty$.
junjios
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CCCC
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1 Answers1

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Making the hints given in the comments more explicit :

cut off $f$ and make a slow transition to $0$

The idea here is to multiply $f$ by a sequence $(u_n)_n$ of smooth compactly supported functions which "act like" $\left(\chi_{[-n,n]}\right)_n$, so that $(f\cdot u_n)_n$ and $\big((f\cdot u_n)'\big)_n$ converge in $L^2(\mathbb R)$ to $f$ and $f'$ respectively.

Such smooth functions are often called "cut-off functions" or "transition functions" and can be explicitly constructed to satisfy a set of desired properties. For instance, using a construction based on mollifiers similar to the ones given here or here, we can construct a sequence of functions $u_n \in C_c^\infty(\mathbb R)$ such that for all $n\ge 1$ $$\begin{cases}& u_n(x) \in [0,1],\ \forall x\in\mathbb R\\ &u_n(x) = 1,\ \forall x\in[-n,n]\\ &u_n(x) = 0,\ \forall x\in\mathbb R\setminus[-2n,2n]\\ &|u_n'(x)|\le 1, \forall x\in\mathbb R \end{cases} $$ Once you have shown (or admitted) that such functions exist, I let you check that the sequences $(f_n := f\cdot u_n)_n$ and $\left(f_n'\right)_n$ respectively converge to $f$ and $f'$ in $L^2(\mathbb R)$.

Now, because $f\in C^1$ and $u_n$ has compact support, it follows that their product $f_n$ is in $C_c^1(\mathbb R)$, and we are done.

Stratos supports the strike
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