**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.**