Question

Hello,

I am very new to the field of approximation theory, and since an extended search on the Internet did not provide answers for two rather basic questions, I decided to ask them here.

1) From my understanding upper bounds for

$$ \inf_{q} \int_{-1}^{1} |f(x) - q(x)|^{2p} dt $$

with $f$ continuous and $q$ a polynomial of degree $n$, are expressed in terms of the $L^p$ smoothness of $f$ and in terms of the degree $n$. Could somebody point me to a proof of such a result?

2) Heuristically, what kind of information do lower bounds for the above infinum contain ? (For example, suppose that I can give a lower bound of $p!$ for the above infinimum as $p \rightarrow \infty$).

My last question might not be well-posed, so if it doesn't make sense please ignore it.

Thank you.

Answer 1

Results on the $L^p$-approximation theory can be found in the basic books on the subject:

Timan, A.F. Theory of approximation of functions of a real variable. Oxford: Pergamon Press. 1963.

Achieser, N.I. Theory of approximation. New York: Frederick Ungar Publishing Co. (1956).

Answer 2

A good introductory lookup for 1) (and similar problems) is the book "Spectral Methods: Fundamentals in Single Domains" by Canuto, Hussaini, Quarteroni & Zang. Chapter 5, in particular. Equation (5.4.16) gives a bound for the $L^p$ norm approximation problem in terms of the L^p smoothness of $f$ and its derivatives: $$ \inf_{q \in \mathbb{P}_n} \| f - q \| _{L^p} \leq C N^{-m} \left ( \sum^{m} _{k=\min(m,n+1)} \| f^{(k)} \|^p _{L^p} \right )^{\frac{1}{p}} $$ According to the bibliographical notes section (p.291) a proof can be found in this paper.