Finding an interval over which a polynomial interpolant $P_n$ converges to a function $f$ as the degree $n$ increases

I understand that due to Runge's phenomenon, increasing the degree, $$n$$, of a polynomial interpolant can actually increase the error between the interpolant, $$P_n$$, and the function, $$f$$, you are trying to approximate, when using equispaced interpolation points.

Therefore I ask the question, is it possible to find some interval $$[-a,a]$$ over which $$P_n$$ does converge to $$f$$ as $$n \to \infty$$, while still using equispaced interpolation points?

In other words, does there exist some $$a$$, such that, $$\underset{-a\leq x\leq a}\max|f(x)-P_n(x)| \to 0$$ as $$n \to \infty$$, for equispaced interpolation points? Where $$f$$ is an arbitrary continuous function $$\mathbb{R} \to \mathbb{R}$$. For example, $$f(x) = cos(\pi x)$$.

If so, how does one compute $$a$$?

• Check out Wikipedia’s page about the Runge’s phenomenon en.m.wikipedia.org/wiki/Runge%27s_phenomenon, where two reasons for it are described. The first is the norm of the derivatives and the second is Lebesque constant. For your example, derivatives are bounded. Then, the estimate for the interpolation error depends on the growth rate of the Lebesque constant and the factorial – VorKir Feb 17 at 18:50