Some time ago, I asked about inite interpolation by a nondecreasing polynomial here at Finite interpolation by a nondecreasing polynomial. This turned out to be an already solved problem; it also turned out that the degree of the solution could not be bounded in terms of the number of interpolation points alone.

My new question is: if we are willing to replace polynomials by another vector space V of indefinitely differentiable functions, then we can we achieve something better than with polynomials, in the sense that V is finite-dimensional?

Formally, fix $x_1 \lt x_2 \lt \ldots \lt x_n$ and let $y_1 \leq y_2\leq \ldots \leq y_n$ vary. We consider the system $(S)$ made of the $n$ interpolation constraints $f(x_i)=y_i$ for $i$ between $1$ and $n$. Is there a finite-dimensional subspace $V$ of ${\cal C}^{\infty}([x_1,x_n],{\mathbb R})$ such that for any $y_1 \leq y_2\leq \ldots \leq y_n$, there is a solution to $(S)$ which is nondecreasing and also in $V$?