Let us first define the variation diminishing property for the Gaussian kernel. Consider a function $f: \mathbb{R} \to \mathbb{R}$ that is sufficiently smooth and define \begin{align} F(x)= \int_{-\infty}^\infty f(t)\, \phi(x-t) {\rm d} t \end{align} where $\phi(t)=\exp(-t^2)$. Then, the variation diminishing property say that \begin{align} S(F) \le S(f) , \end{align} where the quantity $S(g)$ denotes the number of sign changes of a function $g$. Note: This property holds not just for Gaussian kernels but for a large class of kernels called Polya frequency functions.
Question: We are interested in what are some proper ways of generalizing this property to multivariate settings and counting number sign changes of $F$, and whether such generalizations are available in the literature?
More specifically, we are interested in a setting
\begin{align} F( u )= \iint_{\mathbb{R}^2}f({\bf t}) \, \phi({\bf r}(u)-{\bf t}) {\rm d} {\bf t} \end{align} where ${\bf r}(u)$ is path in $\mathbb{R}^2$ where now $\phi ({\bf t})=\exp(-\|{\bf t}\|^2)$. The goal is to provide a bound on the number of sign changes of $F$ using some properties of $f$ and ${\bf r}$. In the univariate case these properties correspond to sign changes of $f$.
Some Thoughts: I have searched the literature and was not able to find any multivariate generalizations. I did, for example, find a generalization to the case where instead of convolution we have more general transformation (ie., $\int_{-\infty}^\infty f(t) k(x,t) {\rm d} t$).
I suspected that the generalization to a full vector case, that is \begin{align} F( {\bf u} )= \iint_{\mathbb{R}^2}f({\bf t}) \phi({\bf u}-{\bf t}) {\rm d} {\bf t} \end{align} is difficult as we need to define a notion of sign changes in $\mathbb{R}^2$. However, in this problem the domain of $F( u )$ is one dimensional, so I think it is a bit easier, and the notion of sign changes is well defined for $F$. The difficulty, however, is how to generalize the notion of sign changes to $f$ or maybe some other property is needed.
