Positivity, monotonicity, and convexity for convolution operators

We consider the convolution inequality \begin{document}$ \begin{equation} a*u {\ge} v\notag \end{equation} $\end{document} for given functions \begin{document}$ a $\end{document} and \begin{document}$ v $\end{document} , and we then investigate conditions on \begin{document}$ a $\end{document} and \begin{document}$ v $\end{document} that force the unknown function \begin{document}$ u $\end{document} to be positive or monotone or convex. We demonstrate that these results for abstract convolution equations can be specialized to yield new insights into the qualitative properties of fractional difference and differential operators. Finally, we apply our results to finite difference methods for fractional differential equations, and we show that our results yield insights into the qualitative behavior of these types of numerical approximations.