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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.

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    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 $x$.2012-06-29

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Ok, so from what I gather I should proceed in the following way:

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 $

et cetera