Polynomial interpolants in quadrature points and L2 convergence spectral rate

Question

We recall that the Lagrange Interpolation Polynomial $p_n(x)$ of a function $f\in C^n(\Omega )$ for some $\Omega \subseteq \mathbb{R}$ and $n\in \mathbb{N}$, has a pointwise error term of the form $$|p_n(x) -f(x)|= \frac{f^{(n)}(\xi (x))}{(n+1)! }\prod\limits_{j=1}^n(x-x_j) \, , $$ where $\{x_j\}\subset\Omega $ are the interpolation points.

This is just one form of the polynomial interpolant of order $n$ through this points. There are many other formulas, that at least theoretically produce the same polynomial.

My question: Given a finite measure $\mu$ on $\Omega$, denote its respective orthogonal polynomial $q_n$ and its roots $x_j ^n$, $j=1,\ldots,N$. What can be generally said about the $L^2 _\mu$ error decay/convergence rate $\|p_n -f\|_2 $ for some smooth enough $f\in H^p _{2,\mu}$, and the Lagrange polynomial in these points?

I'm not sure if it is neccessary or not, but we can limit the discussion to continuous measures $d\mu \ll dx$

What I know- Classical Orthogonal Polynomials: If $\mu$ is a classical measure/ from the Askey scheme, we have spectral $L^2 _{\mu}$ convergence. This is sometime referred to as the polynomial chaos colocation expansion. However, the proof of these results is derived from the spectral properties of the respective orthogonal polynomials, and does not stem directly from the Lagrange interpolation polynomial form. The idea is to use Fourier-like techniques to show that there exists a spectrally convergent polynomial expansion $\Pi _n (f)$, and then show that it actually interpolates $f$ in these points, i.e. that $\Pi _n (x_j ^n) =f(x_j ^n)$. Note that the relevant operator $\Pi _n f$ if not its $L^2$ projection, but its approximation using quadrature formulas.

For an example of the spectral results, see for example Dongbin Xiu, "Numerical methods for stochastic computation", Theorem 3.6.

What else - non Classical: one recent paper shows that if $\nu$ is classical, and $\mu <\nu$ in some normed sense, than the $L^2$ projections spectral convergence is also true for $\mu$, and therefore also the convergence of the polynomial interpolant in $L^2$.

However, Not all measures have this spectral property. If we could have prove something as strong for a general measure directly from the Lagrange interpolation polynomial, that would've been helpful.

Answer 1

It is a classical result that the Lagrange interpolants to $f$ at the zeros of the orthogonal polynomials $q_n$ with respect to an arbitrary measure $\mu$ converge to $f$ in $L^2(\mu)$. For a finite interval $\Omega$, it holds true as soon as $f\in L^2(\mu)$ and has finite moments with respect to $\mu$. This is a result from Erdös-Turan in :

On interpolation. I. Quadrature- and mean-convergence in the Lagrange-interpolation. Ann. of Math. (2) 38 (1937), 142–155.

For an unbounded set $\Omega$, there are some additional growth conditions on $f$ at $\infty$ that still ensure the $L^2(\mu)$ convergence of the Lagrange interpolants, see the paper by Shohat :

On the convergence properties of Lagrange interpolation based on the zeros of orthogonal Tchebycheff polynomials. Ann. of Math. (2) 38 (1937), 758–769.

A proof in the finite interval case can be found on p.333 of Szegö's book on orthogonal polynomials. A short proof in the unbounded case can be found in this paper, see Theorem 11.1.

There are also results (requiring more work) about convergence with respect to $L^p$ norms, $p>1$.

EDIT: Regarding the comments :

For (a): I would also be interested to know if there is an upper bound in terms of derivative for an arbitrary weight $w$ on $[0,1]$. At least, there is the following lower bound about Fourier expansions $S_{n}(f)$ in terms of orthogonal polynomials with respect to $w$, see section 8.6.2 of Timan's book, Theory of approximation of functions of a real variable :

let $C^{r}$ be the set of functions $f$ defined on $[0,1]$ having an $r$-th continuous derivative. Then $$\sup_{f\in C^{r}}\frac{\|f-S_{n}(f)\|_{2,dx}}{\omega(f^{(r)},1/n)}\geq\frac{B_{r}}{n^{r}},$$ where $\omega$ denotes the modulus of continuity and the constant $B_{r}$ depends only on $r$.

For (b): I don't see that the interpolant $p_{n}$ and the Fourier expansion $S_{n}(f)$ coincide. Is it true for a particular weight ?