  In a recent paper, Br\"and\'en, Krasikov, and Shapiro consider root location
preservation properties of finite difference operators. To this end, the
authors describe a natural polynomial convolution operator and conjecture that
it preserves root mesh properties. We prove this conjecture using two methods.
The first develops a novel connection between the additive (Walsh) and
multiplicative (Grace-Szeg\"o) convolutions, which can be generically used to
transfer results from multiplicative to additive. We then use this to transfer
an analogous result, due to Lamprecht, which demonstrates logarithmic root mesh
preservation properties of a certain $q$-multiplicative convolution operator.
The second method proves the result directly using a modification of
Lamprecht's proof of the logarithmic root mesh result. We present his original
argument in a streamlined fashion and then make the appropriate alterations to
apply it to the additive case.
