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Statement

Theorem XIV (pp. 547--548). Suppose that for a matrix

$$|l_\nu(x)|\le D,\qquad-1\le x\le1,\quad\nu=1,\ldots,n,\quad n=1,2,\ldots.$$

Then for the elements $\cos\vartheta_\nu^{(n)}$ ($\nu=1,\ldots,n$) of the $n$th row and every subinterval $[\alpha,\beta]$ of $[0,\pi]$ with $(\beta-\alpha)n\ge c_{69}(D,\epsilon)$,

$$\Bigl|\sum_{\alpha\le\vartheta_\nu^{(n)}\le\beta}1-\frac{\beta-\alpha}{\pi}n\Bigr| <c_{70}(D,\epsilon)\{(\beta-\alpha)n\}^{1/2+\epsilon},$$

where the paper stresses that $c_{70}(D,\epsilon)$ does not depend on $\alpha$ and $\beta$ either.

Proof pointer

Pp. 548--552. Upper estimate: if $[\alpha,\beta]$ holds $k+l$ nodes with $k=[(\beta-\alpha)n/\pi]$, the paper builds the cosine polynomial (76) from the nodes in $[\alpha,\beta]$ and equally spaced points outside, uses Lemma XI to place its maximum outside the interval, multiplies by a transformed Chebyshev polynomial of order $[\frac14l]$ (77), and interpolates at the nodes; comparing with the hypothesis (79)--(84) bounds $l$ by a constant times $(k\log k)^{1/2}$ (cases $l\ge k\ge20$ and $l<k$). Lower estimate (85)--(92): a similar construction with a kernel $\psi$ of the form (88a) shows that $k-l$ nodes with $l>\frac12k^{1/2+\epsilon}$ lead to a contradiction for $l>c_{86}(D,\epsilon)$.

Read depth

Claims checked: Theorem XIV was read clause by clause on the page image of the print; the proof was followed for structure only. Nothing here is independently reviewed.

Dependencies

Lemma XI (stated on the Theorem XII page) of the same paper.

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.