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Statement

Setting (p. 523): the Christoffel numbers of the weight $p$ are $k_\nu^{(n)}=\int_{-1}^1l_{\nu,n}(t)p(t)\,dt$ (display (28)); by (29b) (p. 524) also $k_\nu^{(n)}=\int_{-1}^1l_\nu(t)^2p(t)\,dt$.

Theorem IX (p. 542). Let $p(x)\sqrt{1-x^2}$ be continuous and $p(x)\sqrt{1-x^2}\ge m>0$ in $[-1,1]$. Then, as $n\to\infty$, for every root $x_\nu^{(n)}$ of the $n$th orthogonal polynomial with

$$\quad-\Bigl[1-\frac{\log n}{n^2}\Bigr]^{1/2}\le x_\nu^{(n)}\le \Bigl[1-\frac{\log n}{n^2}\Bigr]^{1/2}$$

one has

$$k_\nu^{(n)}=\int_{-1}^1l_\nu(t)^2p(t)\,dt\sim \frac{\pi p(x_\nu^{(n)})\sqrt{1-x_\nu^{(n)2}}}{n}.$$

Proof pointer

Pp. 539--543. The comparison polynomial $\phi_{n-1}$ of (52) (p. 539), built from Chebyshev polynomials and normalized by $\phi_{n-1}(\xi_0)=1$, minimizes $\int_{-1}^1f^2/\sqrt{1-t^2}$ among polynomials of degree $n-1$ with $f(\xi_0)=1$ (display (51)) and is bounded on $[-1,1]$ (55). Lemma IX (p. 540) evaluates $\lim n\int_{-1}^1\phi_{n-1}^2p$ in terms of $\pi p(\xi_0)\sqrt{1-\xi_0^2}$, and Lemma X (p. 541) shows that a polynomial normalized to $1$ at $\xi_0$ whose weighted square integral near $\xi_0$ falls short of the Chebyshev extremal value must be exponentially large elsewhere. Lemma X and Shohat's minimum property give the lower bound (62)--(63); the minimum property with $\phi_{n-1}$ as competitor gives the upper bound (64).

Read depth

Claims checked: Theorem IX, (28), (29b), (51), (52) and Lemmas IX and X were read clause by clause on the page images of the print; the proof was followed for structure. Nothing here is independently reviewed.

Dependencies

Lemmas II (Shohat's minimum property, Corollary I), IX and X of the same paper; the Christoffel--Darboux formula.

Source. P. Erdős and P. Turán, On interpolation. III. Interpolatory theory of polynomials, Annals of Mathematics (2) 41 (3) (1940), 510--553, DOI 10.2307/1968733; the edition read is named on the source card.

Bears on

None of the problem pages directly.