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Let $\phi \in L^1(\mathbb{R})$ be such that $\int_{-\infty}^\infty \phi(x)dx=1$ and let, for $c>0$, $\phi_c(x)=\frac{1}{\varepsilon} \phi(\frac{x}{c})$.

In the book Introduction to Fourier analysis on Euclidean spaces, by Elías M. Stein,Guido L. Weiss, is the following theorem (Th.1.18):

If $f \in L^p(\mathbb{R})$, where $p\in [1, \infty)$ or in $C_0$ then $\|f*\phi_c-f\|_p \rightarrow 0$ as $c\rightarrow 0^+$.

Proof of this theorem is given there only in the case $p\in [1, \infty)$. How to proof it in the case $f \in C_0 $ ?

($C_0$ is the Banach space, with supremum norm, of all continuous functions $g: \mathbb{R} \rightarrow \mathbb{R}$ such that $\lim_{x \rightarrow +\infty}g(x)=\lim_{x \rightarrow -\infty}g(x)=0$.)


Edit. The mentioned proof in $L^p$ goes in the following way:

$$(f* \phi_c)(x)-f(x)=\int_R [f(x-t)-f(x)]\phi_c(t)dt$$ Next by generalized Minkowski inequality and substitution:

$$\|f* \phi_c-f\|_p \leq \int( \int|f(x-t)-f(x)|^p dx)^\frac{1}{p} \frac{1}{c} |\phi(\frac{t}{c})|dt=\int( \int|f(x-ct)-f(x)|^p dx)^\frac{1}{p} |\phi(t)|dt.$$ Next $\|f(x+h)-f(x)\|_p \rightarrow 0$ when $h \rightarrow 0$ is used and is applied majorized convergence in last integral.

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I suspect that the proof reduces the $L^p$ case to the fact that the space of compactly supported functions $C_c$ is dense in $L^p$. Note that $C_c$ is also dense in $C_0$. – t.b. Apr 4 '12 at 21:53
That's actually much easier isn't it? – Jonas Teuwen Apr 4 '12 at 22:38

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