Although the convolution of the product is not the product of the convolution, i.e. $$fg*h\neq (f*h)(g*h).$$
I am wondering if this true (for a suitable class of functions) in the limit when one uses a delta net.
A delta net is defined as:
A net $\{(\varphi_{n})\}\in(0,1]$ of smooth functions on $\mathbb{R}^{n}$ is called a delta net, if
- $supp( \varphi_{n})\rightarrow {0}$ as $n\rightarrow 0$
- $\int\varphi_{n} d\mu\rightarrow $ 1 as $n\rightarrow 0$
- $\varphi_{n}$ is uniformly bounded in $L^{1}(\Omega,\mu)$
For example if we consider the convolution of $f*\varphi_{n}=f_{n}$ where $f$ is in $L_{loc}^{p}$ and if $g*\varphi_{n}=g_{n}$ is uniformly bounded then: $$\lim_{n\rightarrow 0}||f_{n}g_{n}-fg||_{p_{loc}}\rightarrow 0$$ Of course using the properties of convolutions one also has that: $$\lim_{n\rightarrow 0}||(fg)*\varphi_{n}-fg||_{p_{loc}}\rightarrow 0$$
I am wondering if anyone ones similar results for $L^{p}$ convergence, measure convergence, pointwise convergece, etc with different assumptions for $f$ and $g$.
