\subsection{Variation diminishing matrices}
The theory of variation diminishing matrices is a classical subject, for instance extensively studied in the monograph \cite{karlin1968total}. Variation diminishing properties of a matrix $X \in \Rmn$ are studied via the sign-definiteness of the compound matrices $C_r(X)$, whose $(i,j)-th$ entry is defined by $\det(X_{(I,J)})$, where $I$ is the $i-th$ and $J$ is the $j-th$ element in $\mathcal{I}_{n,r}$ and $\mathcal{I}_{m,r}$, respectively. 
For example, if $X \in \mathbb{R}^{3 \times 3}$, then $C_{[2]}(X)$ reads
\begin{align*}
\begin{pmatrix}
\det(X_{\{1,2 \},\{1,2 \}}) & \det(X_{\{1,2 \},\{1,3\}}) & \det(X_{\{1,2 \},\{2,3\}})\\
\det(X_{\{1,3 \},\{1,2 \}}) & \det(X_{\{1,3 \},\{1,3\}}) & \det(X_{\{1,3 \},\{2,3\}})\\
\det(X_{\{2,3 \},\{1,2 \}}) & \det(X_{\{2,3 \},\{1,3\}}) & \det(X_{\{2,3 \},\{2,3\}})\\
\end{pmatrix}.
\end{align*}
made of all $r$-minors. 
By the \emph{extended Cauchy-Binet formula} \cite{karlin1968total} with $X \in \mathbb{R}^{n \times p}$ and $Y \in \mathbb{R}^{p \times m}$
\begin{equation}
\det((XY)_{I,J}) = \sum_{K \in \mathcal{I}_{p,r}} \det(X_{I,K}) \det(Y_{K,J}), \label{eq:CB}
\end{equation}
where $I \in \mathcal{I}_{n,r}$, $J \in \mathcal{J}_{m,r}$ and $r \leq p$, it easy to verify the following well-known properties \cite[Chapter~6]{fiedler2008special}. 
\begin{lem}\label{lem:compound_mat}
	Let $X \in \mathbb{R}^{n \times p}$, $Y \in \mathbb{R}^{p \times m}$ and $r \in \mathds{Z}_{\geq 1}$.
	\begin{enumerate}[i)]
		\item $C_{[r]}(XY) = C_{[r]}(X)C_{[r]}(Y)$.
		\item $\sigma(C_{[r]}(X)) = \{\prod_{i \in I} \lambda_i(X): I \in \mathcal{I}_{n,r} \}$.
		\item If $X \succeq 0$, then $C_{[r]}(X) \succeq 0$.
	\end{enumerate} 
\end{lem}

\begin{defn}[Sign-consistency and -regularity]
\label{def:sc}
	A matrix is called \emph{ $k$-sign consistent}  if all its  minors of order $k$ have the same sign. The property is {\it strict} if none of those minors is zero. 
The same matrix  is called \emph{(strictly) $k$-sign regular}, if it is (strictly) $j$-sign consistent for all $1 \leq j \leq k$. It is called {\emph (strictly)  $k$-positive} if the sign of minors is the same for all $1 \leq j \leq k$.
\end{defn}

The relationship between the variation diminishing property and the above definitions is provided by combining the results \cite[Theorem~5.1.1.4, Theorem~5.1.1.5,Theorem~5.2.2.4,]{karlin1968total} as follows.
\begin{prop} \label{prop:SR}
	Let $X \in \Rmn$ with $n \geq m$. If $X$ is $\tp{k}$ with $k \leq m$, then 
	\begin{enumerate}[a)]
		\item $\vari{u} \leq k-1 $ implies $\vari{Xu} \leq \min \{\rk(X)-1,\vari{u}\}$ 	 \label{eq:vardim}
		\item $\vari{Xu} = \vari{x}$ implies that the first non-zero element in $Xu$ has the same sign as the first non-zero element in $u$. \label{item:tp_k_order}
	\end{enumerate}
	The converse also holds if $X$ has full column rank. 
\end{prop}
If we disregard the order of sign changes in $Xu$, i.e., $X$ is simply assumed to be $\sr{k}$, then (\ref{prop:SR}) also remains true if (\ref{item:tp_k_order}) is dropped. 

The reader will notice that the number of $k$ minors of $X$ is combinatorial. For our purposes, fortunately, it will be sufficient to check the sign of consecutive minors to verify $k$-positivity \cite{fekete1912uber,fallat2017total}.

	

\begin{prop}\label{prop:con_minor_Hankel_Toep}
	Let $X \in \Rnn$ and $k \leq n$. 
	\begin{enumerate}[i.]
		\item Hankel $X$ is $\tp{k}$ if and only if all consecutive $j$-minors of $X$ are nonnegative for $1 \leq j \leq k$. 
		\item If $X$ is Toeplitz, lower triangular and such that all consecutive $j$-minors of $X_{1:n,1:j}$ are strictly positive for $1 \leq j \leq k-1$ and all consecutive $k$-minors of $X_{1:n,1:k}$ are nonnegative, then $X$ is $\tp{k}$.
	\end{enumerate} 
\end{prop}
Note that the difference between Hankel and Toeplitz matrices esteems from the fact that strictly $\tp{k}$ Hankel matrices lie densely within the set of $\tp{k}$ Hankel matrices \cite{fallat2017total}. 


 


\subsection{Variation-diminishing operators}


In virtue of Proposition~\ref{prop:op_seq}, we extend Definition~\ref{def:sc} to $\Toep{g}$ and $\Hank{g}$ as follows.
\begin{defn}
	$\Toep{g}$ and $\Hank{g}$ are $\sr{k}$ and $\tp{k}$ if the representation matrices of $\Toep{g}^n$ and $\Hank{g}^n$, respectively, are $\sr{k}$ and $\tp{k}$ for all $n \geq 1$. In particular, for $k=\infty$, we refer to them as \emph{totally} sign-regular and positive. Analogously, we call the system $G(z)$ Hankel and Toeplitz $\sr{k}$ and $\tp{k}$.
\end{defn}

The following elementary properties are then inherited from the properties of the finite dimensional compound matrix as reviewed in the Appendix of \cite{grussler2020variation}.
\begin{lem}\label{prop:close}
	Let $\{g_i\}_{i \in \mathds{N}} \subset \linf^{\mathds{N}}$ with $g^\opts = \lim_{i\to\infty} g_i \in \linf^{\mathds{N}}.$
	\begin{enumerate}
		\item Nestedness: $\Toep{g_1}$ is $\sr{k}$ if and only if $\Toep{g_1}$ is $\sr{j}$ for all $1 \leq j \leq k$. The same applies to $\Hank{g_1}$.
		\item Serial interconnection: If $\Toep{g_1}$ and $\Toep{g_2}$ are $\sr{k}$, then $\Toep{g_1 \ast g_2} = \Toep{g_1} \Toep{g_2}$ is $\sr{k}$. 
		\item Parallel interconnection: If $\Hank{g_1}$ and $\Hank{g_2}$ are $\tp{k}$, then $\Hank{g_1 + g_2} = \Hank{g_1}+ \Hank{g_2}$ is $\tp{k}$. 
		\item Closedness: If $\Toep{g_i}$ is $\sr{k}$ for all $i \in \mathds{N}$, then $\Toep{g^\opts}$ is $\sr{k}$. The same applies to $\Hank{g_i}$ and $\Hank{g^\opts}$. 
		\item Multiplication Toeplitz: If $G_1(z)$ is a first order system with $g_1 \in \NF$ and $\Toep{g_2}$ is $\tp{k}$, then $\Toep{g_1 g_2}$ is $\tp{k}$.
	\end{enumerate}
\end{lem}
The analogue of Proposition \ref{prop:SR} for Toeplitz and Hankel operators is then as follows \cite{karlin1968total}:
\begin{prop}\label{prop:sr_inf}
	Let $g \in \lint$ and $\Hank{g}$ be $\sr{k}$ with $k \geq 1$. Then,
	\begin{equation}
	\forall u \in \linf: \ \vari{u} \leq k-1 \ \Rightarrow \ \vari{\Hank{g} u} \leq \vari{u}. \label{eq:vardim_inf}
	\end{equation}
	Conversely, if (\ref{eq:vardim_inf}) holds then $\Hank{g}$  is $\sr{k}$. The same proposition holds for $\Toep{g}$.
\end{prop}
Proposition~\ref{prop:sr_inf} is generally valid for pairs $g$ and $u$ as long as $\Hank{g} u \in \linf$, e.g., $g \in \linf$ and $u \in \lint$. 


In the sequel, we will verify sign-regularity only by inspecting consecutive minors, that is the determinant of the following matrices:
\begin{equation*}
T_g(t,k) := \begin{pmatrix}
g(t) & g(t-1) & \dots & g(t-k+1)\\
g(t+1) & g(t) & \dots & g(t-j)\\
\vdots & \vdots & \ddots   & \vdots \\
g(t+k-1)   & g(t+k-2) & \dots & g(t)\\
\end{pmatrix}
\end{equation*}
and
\begin{equation*}
H_g(t,k) := \begin{pmatrix}
g(t) & g(t+1) & \dots & g(t+k-1)\\
g(t+1) & g(t+2) & \dots & g(t+k)\\
\vdots & \vdots & \ddots   & \vdots \\
g(t+k-1)   & g(t+j) & \dots & g(t-2(k-1))\\
\end{pmatrix},
\end{equation*}
respectively. With a realization $(A,b,c)$, it holds then that
\begin{equation}
H_g(t,j) = \mathcal{O}_j A^{t-1}\mathcal{C}_j \label{eq:Hankel_mat_real}
\end{equation}
for the extended controllability and observability matrices
\begin{subequations}
	\begin{align}
	\mathcal{C}_j &:= \begin{pmatrix}
	b & Ab & \dots & A^{j-1} b
	\end{pmatrix}\\
	\mathcal{O}_j &:= \begin{pmatrix}
	c^\transp & A^\transp c^\transp & \dots & {A^\transp}^{j-1} c^\transp
	\end{pmatrix}^\transp.
	\end{align}
\end{subequations}
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