# For which $n$, can we find a sequence of $n+1$ distinct points s.t. the interpolating polynomial of every +ve continuous function is itself +ve

Fix an interval $$[a,b]$$. For which integers $$n>1$$, does there exist $$n+1$$ distinct points $$\{x_0,x_1,...,x_n\}$$ in $$[a,b]$$ such that for every continuous function $$f:[a,b] \to (0,\infty)$$, the unique interpolating polynomial $$p_n(x)$$ of $$f$$ at the nodes $$\{x_0,x_1,...,x_n\}$$ satisfy $$p_n(x)\ge 0,\forall x\in [a,b]$$ ?

In the present question, we do not want to let the nodes vary with the function.

• This question is really about polynomials only--there is no need to mention any (continuous) functions. – Wlod AA Oct 28 '18 at 6:36

Let $$n \ge 2$$. Given any points $$x_0 < x_1 < \dots < x_n$$, there is a quadratic function positive at all those points, but negative somewhere in $$[x_0,x_n]$$.

Indeed, let $$c \in [x_0,x_n]$$ be any point other than those $$n+1$$ points. There is a quadratic $$\phi(x) = -1+m(x-c)^2$$ that is positive at those points. Simply take $$m$$ large enough.

The function $$f$$ to be approximated has values $$f(x_j) = \phi(x_j)$$, linear between, and constant on $$(-\infty,x_0]$$ and on $$[x_n,\infty)$$. So $$f$$ is continuous on $$(-\infty,\infty)$$ and $$f(x) > 0$$ for all $$x \in (-\infty,\infty)$$.

The interpolating polynomial $$p_n$$ with $$p_n(x_j) = f(x_j)$$ for $$j=0,1,\dots, n$$ is actually $$\phi$$ itself, so $$p_n(c) = -1$$.

For $$n=1$$, take $$x_0,x_1$$ the two endpoints. Your interpolating polynomial is degree $$1$$, the graph is a straight line, so if it is positive at the endpoints, then it is also positive at all points between.

• and for $n=2$ ... ? – user521337 Oct 27 '18 at 0:49
• For $n=2$, you need nodes $x_0,x_1,x_2$ ... also, is your $f(x)$ positive only at the nodes ? If it so, then you've got it wrong, I want $f:[a,b]\to (0,\infty)$, that means $f(x)>0,\forall x\in [a,b]$ – user521337 Oct 27 '18 at 16:23
• OK, $n+1$ nodes means the approximation has degree $n$ Fixing. – Gerald Edgar Oct 27 '18 at 17:34