text: # Problem from Brezis's book (mollifiers)
  
  Any ideas on how to get started with this?
  
  Let $\rho \in L^1(\mathbb{R}^N)$ with $\int \rho=1$. Set $\rho_n(x)=n^N \rho(nx)$. Let $f \in L^p(\mathbb{R}^N)$. Show that $\rho_n \star f \to f$ in $L^p(\mathbb{R}^N)$.
  
  The proof of Theorem 4.22 would almost go through for this case. The problem is that I don't know anything about the support of the $\rho_n$'s.This seems to be crucial for the proof of Proposition 4.21 which is used to prove the theorem.
  
  -
  Could you give a link to google book's or write down the cited theorems? – martini Jun 29 '12 at 14:34
  It's Exercise (4.28), on page 127, of the Springer edition. Just found it. – Siminore Jun 29 '12 at 14:39
  You know that $\int_{\mathbb{R}^n\backslash B_R(0)} \rho$ becomes arbitrily small if $R$ is sufficiently large, that should suffice. I don't think this is true if $p=\infty$. – user20266 Jun 29 '12 at 14:42
  You can prove that $$\text{support}(\rho_n)=\frac{1}{n}\text{support}(\rho).$$ – Mercy King Jun 29 '12 at 14:53
  I don't know your approach of proof, but you should note that, unlike in the case of smooth $\rho$ with compact support, you will only get convergence in $L^p$, $1 \le p < \infty$. In particular, as pointed out by an example in Stein's book on Harmonic analysis (Chapter II, § 5.16), $\lim_{n\rightarrow\infty} \rho_n \star f(x)$ may fail to exist for almost every $x$. – user20266 Jun 29 '12 at 15:45
  
  Define $$\bar \rho = \frac{I_{B(R)} \: \rho}{\|I_{B(R)} \: \rho\|_1},$$ where $R>0$ is such that $\int_{\mathbb{R}^N \setminus B(R)} \rho < \epsilon$. Then I write
  $$\| \rho_n \star f - \bar \rho_n \star f + \bar \rho_n \star f -f \|_p \leq \| \rho_n \star f - \bar \rho_n \star f \|_p + \| \bar \rho_n \star f -f \|_p$$ $$\leq \| \rho_n - \bar \rho_n\|_1 \|f\|_p + \| \bar \rho_n \star f -f \|_p$$
explanation: The document is a discussion about a problem from Brezis's book on mollifiers. It contains a math question and several comments and suggestions from different users on how to approach the problem. The main question is about showing that \(\rho_n \star f \to f\) in \(L^p(\mathbb{R}^N)\) for a given function \(\rho\) and sequence \(\rho_n\).
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
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  1. Briefly describe the document and mention if it contains a question or a question and answer.
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distilabel_metadata.role: system
distilabel_metadata.content: # Problem from Brezis's book (mollifiers)
  
  Any ideas on how to get started with this?
  
  Let $\rho \in L^1(\mathbb{R}^N)$ with $\int \rho=1$. Set $\rho_n(x)=n^N \rho(nx)$. Let $f \in L^p(\mathbb{R}^N)$. Show that $\rho_n \star f \to f$ in $L^p(\mathbb{R}^N)$.
  
  The proof of Theorem 4.22 would almost go through for this case. The problem is that I don't know anything about the support of the $\rho_n$'s.This seems to be crucial for the proof of Proposition 4.21 which is used to prove the theorem.
  
  -
  Could you give a link to google book's or write down the cited theorems? – martini Jun 29 '12 at 14:34
  It's Exercise (4.28), on page 127, of the Springer edition. Just found it. – Siminore Jun 29 '12 at 14:39
  You know that $\int_{\mathbb{R}^n\backslash B_R(0)} \rho$ becomes arbitrily small if $R$ is sufficiently large, that should suffice. I don't think this is true if $p=\infty$. – user20266 Jun 29 '12 at 14:42
  You can prove that $$\text{support}(\rho_n)=\frac{1}{n}\text{support}(\rho).$$ – Mercy King Jun 29 '12 at 14:53
  I don't know your approach of proof, but you should note that, unlike in the case of smooth $\rho$ with compact support, you will only get convergence in $L^p$, $1 \le p < \infty$. In particular, as pointed out by an example in Stein's book on Harmonic analysis (Chapter II, § 5.16), $\lim_{n\rightarrow\infty} \rho_n \star f(x)$ may fail to exist for almost every $x$. – user20266 Jun 29 '12 at 15:45
  
  Define $$\bar \rho = \frac{I_{B(R)} \: \rho}{\|I_{B(R)} \: \rho\|_1},$$ where $R>0$ is such that $\int_{\mathbb{R}^N \setminus B(R)} \rho < \epsilon$. Then I write
  $$\| \rho_n \star f - \bar \rho_n \star f + \bar \rho_n \star f -f \|_p \leq \| \rho_n \star f - \bar \rho_n \star f \|_p + \| \bar \rho_n \star f -f \|_p$$ $$\leq \| \rho_n - \bar \rho_n\|_1 \|f\|_p + \| \bar \rho_n \star f -f \|_p$$
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a discussion about a problem from Brezis's book on mollifiers. It contains a math question and several comments and suggestions from different users on how to approach the problem. The main question is about showing that \(\rho_n \star f \to f\) in \(L^p(\mathbb{R}^N)\) for a given function \(\rho\) and sequence \(\rho_n\).
  
  Label: question-and-answer