Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
    
It is unfortunate that you let $p$ denote both a polynomial and a positive integer. I guess that it is clear that you take the infimum over polynomials, of a certain degree? But this should be clarified. –  Johan Öinert Nov 13 '11 at 1:35
    
Can you clarify 2) please? Wouldn't it make more sense to study lower bounds as $n\to\infty$ when $f$ is taken to be the worst one from some function space? –  timur Nov 30 '11 at 23:55
add comment

2 Answers

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.

share|improve this answer
add comment

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).

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.