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

Hi everybody, this is my first question.$L^2$,$H^p$ are the standard Lebesgue,Sobolev spaces here, and I am deliberately omitting the domains because I'll accept an answer if it's on an interval or a more complicated domain.

A cornerstone result in constructive approximation theory is the derivation of the "Jackson inequality," which is effectively shows that the operator norm of the residual operator of $L^2$ projection onto degree $N$ polynomials is bounded. More specifically, take $P^N$ to be the space of polynomials of degree not exceeding $N$, $\mathcal{P}^N : L^2 \to P^N$ to be the standard $L^2$ projection operator. Then the Jackson inequality seeks to calculate

$$ ||I - \mathcal{P}^N|| = \sup_{||u||_{H^p} = 1}||(I - \mathcal{P}^N)u|| $$

and usually it has some concrete dependence on order $N$ and regularity $p,$ but maybe an unknown constant that does not depend on these parameters.

I was wondering if there are similar results in taking a generic sequence of subspaces instead of $P^N$ ? I'm curious if there is a known connection between properties of the basis (say: Bernstein inequalities) and the size of the corresponding Jackson inequality.

share|improve this question
add comment

Your Answer

 
discard

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

Browse other questions tagged or ask your own question.