Integral lower bound for the Lebesgue function

---

Statement

Setting (p. 191). For each $n$ the nodes are

$$-1\le x_{1,n}<x_{2,n}<\cdots<x_{n,n}\le1, \tag{1}$$

written $x_k=x_{k,n}$; with $\omega(x)=\prod_{k=1}^n(x-x_k)$ the fundamental polynomials are

$$l_k(x)=\frac{\omega(x)}{\omega'(x_k)(x-x_k)}\qquad(k=1,\ldots,n),$$

and for $-1\le a<b\le1$

$$\lambda_n(a,b)=\max_{a\le x\le b}\sum_{k=1}^n|l_k(x)|.$$

Theorem (p. 191, unnumbered, displayed as (4)). For every system of nodes (1) and every subinterval $[a,b]$ of $[-1,1]$,

$$\int_a^b\sum_{k=1}^n|l_k(x)|\,dx\ge c_3(b-a)\log n \qquad(n\ge n_2(a,b)). \tag{4}$$

Here $c_3$ is an absolute positive constant (footnote 1, p. 191: the lettered constants $c_1,c_2,\ldots$ are absolute positive constants) and the threshold is written $n_2(a,b)$, a function of the interval alone, so it does not depend on the nodes. The paper gives no value of $c_3$ in the statement; its last display (p. 195) gives $(b-a)\log n/40$.

The paper says (p. 191) that Bernstein's local bound (3), $\lambda_n(a,b)\ge c_2\log n$ for $n\ge n_1(a,b)$, follows from the theorem as a corollary, and that the case $a=-1$, $b=1$ was announced in P. Erdős, Problems and results on the theory of interpolation. II, Acta Math. Acad. Sci. Hungar. 12 (1961), 235--244. The corollary is immediate: the integral in (4) is at most $(b-a)\lambda_n(a,b)$, so (4) gives $\lambda_n(a,b)\ge c_3\log n$ for $n\ge n_2(a,b)$.

Closing remark (p. 195): "The best constants in (2) and (3) are (roughly speaking) $2/\pi$." The authors add that their $c_3$ is apparently far from best possible and that their method does not seem suited to finding the largest $c_3$.

Source. P. Erdős and J. Szabados, On the integral of the Lebesgue function of interpolation, Acta Math. Acad. Sci. Hungar. 32 (1--2) (1978), 191--195: the setting and the theorem on p. 191, Case 1 on p. 192, the [[polynomials/erdos_szabados_1978_integral_lebesgue_function_interpolation/node_gap_lemma|node gap lemma]] on pp. 192--193, Case 2 on pp. 192--195. The edition read is identified on the [[polynomials/erdos_szabados_1978_integral_lebesgue_function_interpolation/_index|source card]].

Read depth. Claims checked: the statement, its quantifiers and the closing remark were read clause by clause on the printed pages. Proof verified, conditional on three external inputs (see Dependencies): the printed argument was read step by step; its late part has the defects listed below, and the theorem with $c_3=1/256$ is proved by the two companions named there, each separately reviewed. The composed chain, with this page and both companions as they stood on 2026-09-18T07:24:04Z, passed the fresh full-chain review, graded PASS for contract and independence by its distinct grade on 2026-09-18. The earlier full-chain review of 2026-09-06 is retained but void as an independent warrant for the composition, after a 2026-09-18 ruling that its delivered candidates carried the companions' earlier-review verdicts. No review credits the printed $1/40$ or the coefficient $2/\pi$.

Proof pointer

Pages 192--195, in two cases.

Case 1, $\lambda_n(a,b)\ge n^3$ (p. 192). At a point where the maximum is attained, a signed sum of the $l_k$ is a polynomial of degree less than $n$ that equals the maximum there and is dominated by $\sum_k|l_k|$ on $[a,b]$. Markov's inequality keeps it above half the maximum on an interval of length of order $(b-a)/n^2$, so the integral is at least of order $(b-a)n$, more than (4) needs.

Case 2, $\lambda_n(a,b)<n^3$ (pp. 192--195). The node gap lemma makes every gap between consecutive nodes in $[a,b]$ at most $75\log n/n$. The integral is bounded below by the contributions of adjacent pairs $|l_k|+|l_{k+1}|$ over the gaps between consecutive nodes in $[a,b]$. For two such gaps, an affine map between them and the Erdős--Turán inequality $l_k(y)+l_{k+1}(y)\ge1$ on $[x_k,x_{k+1}]$ give a lower bound for each pair; adding a pair to its mirror image removes the ratio of $\omega$ values. This leaves, up to an absolute factor, a double sum of $\Delta x_m\,\Delta x_k/(x_{k+1}-x_m)$ with $\Delta x_k=x_{k+1}-x_k$, displayed as (8) (p. 194). For each gap in the left half of $[a,b]$, grouping the later gaps into blocks of length $75\log n/n$ makes the inner sum at least a harmonic sum of order $\log n$ (p. 195), and the outer sum of the $\Delta x_m$ is of order $b-a$.

Defects in the printed late argument (an observation of this page).

1. In both (7) and (8) the triangular sum is printed with $k=m$ allowed. The preceding half-sum over all pairs contains each diagonal term with weight $1/2$, whereas the triangular symmetrization counts it in full; and the ratio computation on p. 194 assumes two distinct ordered gaps, so it does not cover the diagonal.
2. The claim that the first and last nodes in $[a,b]$ tend to $a$ and $b$ is justified only parenthetically (p. 193). The intervals $I_{t,m}$ of p. 194 can extend beyond $b$ for the largest $t$ in the displayed range; the direct bound for the denominators is $(t+2)75\log n/n$, not the printed $(t+1)75\log n/n$; and the final grouping by three consecutive intervals (p. 195) is not written as a disjoint selection.

A compilation-authored [[polynomials/erdos_szabados_1978_integral_lebesgue_function_interpolation/finite_symmetrization_correction|finite symmetrization correction]] treats the diagonal and off-diagonal terms separately and proves the analog of (8) with prefactor $1/16$ in place of $1/8$; it passed the diagonal review, whose scope is that finite step only. A compilation-authored [[polynomials/erdos_szabados_1978_integral_lebesgue_function_interpolation/endpoint_harmonic_completion|endpoint and harmonic-block completion]] proves the end-gap control, uses disjoint blocks and the shifted denominators, includes Case 1, and concludes (4) with $c_3=1/256$ under an explicit threshold; it passed the late-proof review without author changes. Neither companion is text of the paper or an author's erratum. The printed $1/40$ is not verified here; a repaired proof may use a smaller absolute constant without changing (4).

Dependencies

• The same paper's [[polynomials/erdos_szabados_1978_integral_lebesgue_function_interpolation/node_gap_lemma|node gap lemma]] (5), for Case 2.
• Bernstein's local bound (3), quoted on p. 191 from S. Bernstein, Sur la limitation des valeurs d'un polynome, Bull. Acad. Sci. de l'URSS 8 (1931), 1025--1050, through the node gap lemma.
• Markov's inequality for the derivative of a polynomial on an interval, used in Case 1; the paper gives no reference for it.
• The adjacent-polynomial inequality $l_k(y)+l_{k+1}(y)\ge1$ for $x_k\le y\le x_{k+1}$, cited on p. 194 as Lemma IV of P. Erdős and P. Turán, On interpolation. III, Ann. of Math. 41 (1940), 510--552; see [[polynomials/erdos_turan_1940_on_interpolation_iii/lemma_iv_adjacent_fundamental_polynomials|Lemma IV and its increasing-node form]].

The proofs of Bernstein's bound and Markov's inequality are outside the reviews recorded here; the Erdős--Turán input has its own reviewed reconstruction.

Bears on

• Problem 1153: the problem asks whether, for every fixed $-1\le a<b\le1$, $\max_{[a,b]}\lambda>(2/\pi-o(1))\log n$. Theorem (4) gives, through the corollary above, $\max_{[a,b]}\lambda\ge c_3\log n$ for $n\ge n_2(a,b)$ with an unspecified absolute $c_3>0$, which is the logarithmic order of the question without its coefficient $2/\pi$. The paper does not prove the bound with coefficient $2/\pi$, and says its method does not seem suited to finding the best constant.