Take the 2-minute tour ×  Let $f,g \in L^2(\mathbb{R^n})$, $\{
f_n \}, \{ g_m \} \subset C^\infty_0(\mathbb{R}^n)$ (infinitely
differentiable functions with compact support) where $f_n \to f$
in $L^2$, and $g_n \to g$ in $L^2$.  Then $f_n \star g_m(x) \to
f \star g(x)$ pointwise as $n,m \to \infty$ (where $\star$
denotes convolution).  Is it true that $\int_{\mathbb{R}^n} f_n
\star g_m dx \to \int_{\mathbb{R}^n} f \star g dx$?  I've
thought about using the dominated convergence theorem, but I
couldn't figure out how, since I don't really know what I can do
with the sequences of functions given only the facts at hand.
share|improve this question add comment  2 Answers  up vote 5
down vote accepted  No it is not true.  First note that we can
improve pointwise convergence to uniform convergence of $f_{n}
\ast g_{m} \to f \ast g$. This is an easy consequence of
Hölder's inequality (markup doesn't like my subscripts, so I'll
use $\|\cdot\|$ for the $L^{2}$-norm): \[ |(f \ast g)(x)| \leq
\int |f(x - y) g(y)|\,dy \