I have a quick question about certain algebraic properties of convolution. If I have 3 functions $f(x)$, $g(x)$ and $h(x)$, is the following true?
$\Big[ f(x) \cdot g(x)\Big] \circ h(x) = \Big[f(x) \circ h(x)\Big] \cdot g(x)$
Where "$\cdot$" and "$\circ$" are multiplication and convolution respectively.
Did you try writing out these products? The first expression is $$[(fg) \circ h](x) = \int f(y) g(y) h(x-y) \, dy.$$ The second is $$ [(f \circ h) g](x) = \left(\int f(y) h(x-y) \, dy \right) g(x).$$
These are clearly not the same. For instance, if $h$ is constant the first expression is indepedent of $x$.