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.