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Statement

Setting (p. 511). For a weight $p(x)\ge0$, Lebesgue-integrable on $[-1,1]$, $\omega_n(x)$ is the polynomial of degree $n$ with leading coefficient $1$ orthogonal to all $\omega_m$, $m\ne n$, with respect to $p$ (displays (8a), (8b)); its roots form the $p$-matrix (p. 512).

Theorem II (p. 522). Let the $L$-integrable weight $p(x)$ be non-negative in $[-1,1]$ and satisfy $p(x)\ge m>0$ on the subinterval $[a,b]$. Then for $a\le x\le b$

$$|\omega_n(x)|\le\left[\frac{8}{(b-a)m}\int_{-1}^1p(t)\,dt\right]^{1/2} \cdot\frac{2n+1}{2^n},$$

and for $a+\epsilon\le x\le b-\epsilon$

$$|\omega_n(x)|\le2\left[\frac{1}{m[\epsilon(b-a-\epsilon)]^{1/2}} \int_{-1}^1p(t)\,dt\right]^{1/2}\cdot\frac{\sqrt{2n+1}}{2^n}.$$

The introduction states the same two bounds as (18a) and (18b) (p. 516) with other numerical constants ($72$ and $12$), with the factors $n$ and $\sqrt n$, with $[\epsilon(b-a)]^{1/2}$ in the second, and for $n=1,2,\ldots$; the constants above are the theorem's own. The paper calls it probable (pp. 516 and 523) that the factor $\sqrt n$ can be replaced by a constant $c_{14}(\epsilon,a,b,m)$, supported by the mean-square bound (27), and does not prove it.

Proof pointer

P. 523. Since $\omega_n$ minimizes $\int_{-1}^1f^2p$ among monic polynomials of degree $n$, comparison with the monic Chebyshev polynomial gives (26), $m\int_a^b\omega_n^2\le4\cdot2^{-2n}\int_{-1}^1p$; Markov's inequality (first bound) or the Bernstein--Fejér inequality (second bound) turns this into a pointwise bound. Pp. 523--527 give a second, interpolatory proof through Lemma II (p. 524), the monotonicity in the weight of $\sum_\nu l_\nu(x_0)^2/k_\nu$ where the $k_\nu$ are the Christoffel numbers (28), with corollaries (34a), (34b) and (35); it yields the same shape with other constants, and the bound (36) (p. 527) of order $\sqrt n/2^n$ on all of $[a,b]$ when $p(x)\ge m/[(x-a)(b-x)]^{1/2}$ there.

Read depth

Claims checked: Theorem II, (18a), (18b), (26), (27) and (36) were read clause by clause on the page images of the print; both proofs were followed for structure. Nothing here is independently reviewed.

Dependencies

None in the corpus. External inputs named by the paper: the minimum property of orthogonal polynomials, Markov's inequality and the Bernstein--Fejér inequality for derivatives of 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.