Does this type of real functions have a name ? What are their characteristics? Let $n > 0$ be an integer. Let $f:\mathbb R^+ \times \mathbb R^+ \rightarrow \mathbb R_0$ be a symmetric function such that the $(n+1) \times k$ matrix
\begin{equation}
\mathbb M=
\begin{pmatrix}
f(s_{k-n+1},s_1) & \ldots & f(s_{k-n+1},s_k)\\
\vdots &  & \vdots \\
f(s_{k+1},s_1) & \ldots & f(s_{k+1},s_k)\\
\end{pmatrix}
\end{equation}
has at most $n$ linearly independent rows for any $k>n$ and real sequence $s_1<s_2<\ldots<s_{k+1}$. What does that say about $f$ ? Do such function have a name ? What are their characteristics/properties ? Is there an easier "definition" that would be equivalent, without matrices for example? Or a stronger "easier" property that would imply the above property ?
 A: This is a special case, but it is also a broad class. It isthe exact solution if $n=1$. If you take $f(x, y) = h(\min \{ x, y\} ) g(\max \{x,y\} ) $, then the first $(k-n+1) $ columns are multiple of $( g(s_{k+n-1}), \ldots , g(s_{k+1}) ) $ . The rank remain then unchanged if we discard the first $k-n$ columns; now we have $k-(k-n) = n $ columns, so that the rank is at most $n$ as desired. Note also that such functions are symmetric.
Conversely, suppose $n=1$ and set $k=2$. Taking determinant one obtains (the function is never zero)
$$ \frac{ f(s_2, s_2) }{ f(s_3, s_2) } = \frac{ f(s_2, s_1) }{f(s_3, s_1) } $$
Which means that the quotient on the right does not depend on $s_1 < s_2$. Taking $s_3 = L$ very big we get that for all $s_1 < s_2 < L$ :
$$ f(s_2, s_1) = \frac{ f(s_2, s_2) }{f(L, s_2) } f(L, s_1) = g(s_2) \cdot h(s_1) $$
By symmetry we will have that for $s_2 < s_1 < L$
$$ f(s_1, s_2) = f(s_2, s_1) = g(s_1) h(s_2) $$
So that $f(x, y) = h(\min \{x, y\}) g(\max \{x, y\}) $. We can repeat this argument for greater $L$, but since the new $h, g$ must agree with the old ones on $s_1 , s_2 < L$, we will get well defined functions $h, g$ on all of $\mathbb{R}^+$. Hope you enjoy!
