Dear Researchers,

Let $\left[\alpha,\beta\right]$ be a non-empty interval and consider an $\left(n+1\right)$-dimensional subspace $\mathbb{S}_{n}$ of $C^n\left(\left[\alpha,\beta\right]\right)$ such that $\mathbb{S}_{n}$ also has a strictly totally positive normalized basis \begin{align} \mathcal{B}_n:=\left\{b_{n,i}\left(u\right): u \in \left[\alpha, \beta\right]\right\}_{i=0}^{n} \end{align} that is formed by unimodal basis functions, i.e., the row-stochastic matrix \begin{align} B:=\left[b_{n,i}\left(u_j\right)\right]_{j=0,\,i=0}^{n,\,n} &:= \left[ \begin{array}{cccc} b_{n,0}\left(u_0\right) & b_{n,1}\left(u_0\right) & \cdots & b_{n,n}\left(u_0\right) \\ b_{n,0}\left(u_1\right) & b_{n,1}\left(u_1\right) & \cdots & b_{n,n}\left(u_1\right) \\ \vdots & \vdots & \ddots & \vdots \\ b_{n,0}\left(u_n\right) & b_{n,1}\left(u_n\right) & \cdots & b_{n,n}\left(u_n\right) \end{array} \right] \end{align} is strictly totally positive for any sequence \begin{align} \alpha \leq u_0 < u_1 < \ldots < u_n \leq \beta \end{align} of strictly increasing knot values $\left\{u_j\right\}_{j=0}^n$ and there exist unique parameter values (maximum points) $\left\{ \overline{u}_i \right\}_{i=0}^n \subset \left[\alpha,\beta\right]$ for which \begin{align} b_{n,i}^{\left( 1\right) }\left( \overline{u}_{i}\right) &=0,\\ b_{n,i}^{\left( 2\right) }\left( \overline{u}_{i}\right) &<0,\\ b_{n,i}^{\left(1\right) }\left( u\right) &>0,~\forall u\in\left[ \alpha,\overline{u}_{i}\right),\\ b_{n,i}^{\left( 1\right) }\left( u\right) &<0,~\forall u\in\left(\overline{u}_{i},\beta\right] \end{align} for all $i=0,1,\ldots,n$.

For example, if one considers the vector space \begin{align} \mathbb{S}_n = \left\langle 1,u,\ldots,u^n : u \in \left[\alpha,\beta\right]\right\rangle \end{align} of polynomials of degree at most $n$, then its unique strictly totally positive normalized basis is formed by the Bernstein polynomials \begin{align} b_{n,i}\left(u\right) = \binom{n}{i} \left(\frac{u - \alpha}{\beta - \alpha}\right)^i \left(1-\frac{u - \alpha}{\beta - \alpha}\right)^{n-i},~u\in\left[\alpha,\beta\right] \end{align} of degree $n$ and $\overline{u}_i = \alpha + \frac{i}{n} \cdot \left(\beta - \alpha\right)$ for all $i=0,1,\ldots,n$.

Note that, in non-polynomial or mixed vector spaces of functions, in general, we do not know the explicit analytical form of the basis functions $\left\{b_{n,i}\right\}_{i=0}^n$.

Consider now a sufficiently smooth non-negative kernel function $\rho\left(u;\sigma\right)$, $u \in \mathbb{R}$ of shape parameter $\sigma \geq 0$ and based on the maximum points $\left\{\overline{u}_i\right\}_{i=0}^{n}$ define the system \begin{align} \mathcal{R}_n^{\sigma}:=\left\{\rho_{n,i}\left(u;\sigma\right):=\rho\left(u-\overline{u}_i; \sigma\right): u \in \left[\alpha,\beta\right]\right\}_{i=0}^n \end{align} of radial basis functions such that:

• the length of the support $\operatorname{supp}\left(\rho\right)=\overline{\left\{u \in \mathbb{R} : \rho\left(u;\sigma\right) \neq 0\right\}}$ is at least $2\left(\beta - \alpha\right)$;
• $\rho$ is a bell-shaped unimodal even function;
• and, concerning the shape parameter $\sigma \geq 0$, one also has the properties: \begin{align*} \rho\left(u; 0\right) &= 1,~ \forall u \in \mathbb{R},\\ \\ \lim_{\sigma \to \infty }\rho\left(u; 0\right) &= \left\{\begin{array}{ll}0,&u \neq 0,\\ 1,&u = 0.\end{array}\right. \end{align*}

For example, one may consider the Gaussian kernel function $\rho\left(u;\sigma\right) = e^{-\left(\sigma u\right)^2}$, $u \in \mathbb{R}$, $\sigma \geq 0$, but my question below is independent of the type of the applied kernel function.

Question: what further conditions (if any) of the (parent) kernel function $\rho$ and of its shape parameter $\sigma$ ensure the strictly total positivity of the perturbed function system \begin{align} \widetilde{\mathcal{B}}_n:=\mathcal{B}_n \circ \mathcal{R}_n^{\sigma}:=\left\{\widetilde{b}_{n,i}\left(u;\sigma\right):= b_{n,i}\left(u\right) \cdot \rho_{n,i}\left(u; \sigma\right) : u \in \left[\alpha,\beta\right] \right\}_{i=0}^{n}, \end{align} i.e., when is strictly totally positive the Hadamard (entry-wise) product of matrices $B$ and \begin{align} R_{\sigma}:=\left[\rho_{n,i}\left(u_j;\sigma\right)\right]_{j=0,\,i=0}^{n,\,n} &:= \left[ \begin{array}{cccc} \rho_{n,0}\left(u_0;\sigma\right) & \rho_{n,1}\left(u_0;\sigma\right) & \cdots & \rho_{n,n}\left(u_0;\sigma\right) \\ \rho_{n,0}\left(u_1;\sigma\right) & \rho_{n,1}\left(u_1;\sigma\right) & \cdots & \rho_{n,n}\left(u_1;\sigma\right) \\ \vdots & \vdots & \ddots & \vdots \\ \rho_{n,0}\left(u_n;\sigma\right) & \rho_{n,1}\left(u_n;\sigma\right) & \cdots & \rho_{n,n}\left(u_n;\sigma\right) \end{array} \right] \end{align} for any strictly increasing knot values $\left\{u_j\right\}_{j=0}^n \subset \left[\alpha,\beta\right]$? Is this even possible at least for a class of kernel functions?

Thank you for your time and energy for dealing with my question.