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Statement

Setting as on the Theorem II page: $\omega_n$ is the monic orthogonal polynomial of degree $n$ of the weight $p$ (p. 511).

Theorem III (p. 528). Let the weight $p(x)$ be non-negative and $L$-integrable in $[-1,1]$, and suppose $p(x)\ge m>0$ throughout the subinterval $[a,b]$. If $x_d^{(n)}$ denotes the root of $\omega_n(x)$ nearest to $x$, then for real $x$

$$|\omega_n(x)|\ge\left[\frac{c_{34}\,m}{(b-a)\int_{-1}^1p(t)\,dt}\right]^{1/2} \left(\frac{b-a}{4}\right)^n|x-x_d^{(n)}|,$$

where, as everywhere in the paper (p. 512), $c_{34}$ is a positive constant independent of $x$ and $n$. The introduction announces it as (19a) (p. 516) with a constant $c_{11}$.

Proof pointer

Pp. 528--530. Lemma III (p. 528): for weights $p_1\ge p_2\ge0$ on $[-1,1]$, both $L$-integrable, with orthogonal polynomials $\omega_n$, $\omega_n^+$, Christoffel numbers $k_\nu$, $k_\nu^+$ and nodes $x_\nu$, $x_\nu^+$, $\sum_\nu1/(k_\nu\omega_n'(x_\nu)^2)\le\sum_\nu1/(k_\nu^+\omega_n^{+\prime}(x_\nu^+)^2)$. Taking $p_2=m$ on $[a,b]$ and $0$ elsewhere gives (38) (p. 529), $\sum_\nu1/(k_\nu\omega_n'(x_\nu)^2)\le\frac1m(2n-1)\binom{2n-2}{n-1}^2(b-a)^{-(2n-1)}$. The interpolation identity (39) expresses $\omega_n(x)^2$ through $\sum_\nu l_\nu(x)^2/k_\nu$; for $x$ between two adjacent roots, Lemma IV gives $l_d(x)^2+l_{d+1}(x)^2\ge\frac12$, and $k_\nu<\int_{-1}^1p$ with the asymptotics of the binomial coefficient finishes the proof (p. 530).

Read depth

Claims checked: Theorem III, Lemma III and (38) were read clause by clause on the page images of the print; the proof was followed for structure. Nothing here is independently reviewed.

Dependencies

Lemma IV of the same paper; Lemma III of the same paper; the minimum property of orthogonal polynomials.

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.