Most of error analysis of polynomial interpolation employs $L_{\infty}$ norm. However, $||\prod\limits_{i=1}^{n}(\cdot-x_i)||_{L^{\infty}(a,b)}\leq\frac{n!h^{n+1}}{4} $ when equidistant points are used, i.e., $x_i=a+ih$ with $h=(b-a)/n$. The bound $\frac{n!h^{n+1}}{4}$ is not very small compared that in Chebyshev nodes. My question is that is it possible to reduce the bound for equidistant points when applying $||\prod\limits_{i=1}^{n}(\cdot-x_i)||_{L^{p}}$ for some $p<\infty$?
Asked
Active
Viewed 189 times
0
-
I think the answer is actually no, at least for small p. The whole problem becomes confined to a small vicinity of the endpoints similar to the Gibbs phenomenon in Fourier series. – Ian May 21 '18 at 15:09
-
For small $p$ one can reduce the bound by (smoothly) increasing the density of interpolation points as the endpoints are approached. But then, that violates the condition of "equidistant points". – Mark Fischler May 21 '18 at 15:17
-
Er, sorry, I misread the question, I meant to say that I think the answer is actually yes for small $p$, in the sense that the bound for small $p$ is better than simply naively plugging in the bound for $p=\infty$ in and integrating. Again the reason is that the problem is highly concentrated near the endpoints, and for small $p$ localized large errors are given less weight compared to large $p$. – Ian May 21 '18 at 15:24
-
Thanks for your replies. I have found some useful results on the book "Error Inequalities in Polynomial Interpolation and Their Applications" by Ravi P. Agarwal and Patricia 1. Y. Wong. – Lin Xuelei Jun 01 '18 at 04:45