Deboor 1978 conjectures bernstein erdos optimal nodes polynomial interpolation

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Carl de Boor and Allan Pinkus, Proof of the conjectures of Bernstein and Erdős concerning the optimal nodes for polynomial interpolation, J. Approx. Theory 24 (1978), 289--303; DOI 10.1016/0021-9045(78)90014-X. Received 1 April 1977.

The copy read for this card is a scan of the journal version, the fifteen printed pages 289--303 (physical PDF p. $n$ is printed p. $288+n$) with an OCR text layer whose formulas are garbled, so the statements below were checked on the page images. Provenance: downloaded in September 2026; the download URL was not recorded; 679,346 bytes. The journal version is the only version read. The scan prints "Copyright © 1978 by Academic Press, Inc. All rights of reproduction in any form reserved." on its first page, every other right reserved.

Reading depth is claims checked for the two conjectures as stated on p. 290, Kilgore's theorem as restated on p. 291, and Theorems 1 (p. 295), 2 (p. 298) and 3 (p. 301), read clause by clause on the page images. The proofs were read for their structure only.

Contents

• Setting (pp. 289--290): fix $n\ge2$ and nodes $a=t_0<t_1<\cdots<t_n=b$ in $[a,b]$; $T$ is the set of such node vectors $\mathbf t$. $P_{\mathbf t}$ is Lagrange interpolation at the $n+1$ nodes, with $\|P_{\mathbf t}\|=\|\Lambda_{\mathbf t}\|_\infty$ for the Lebesgue function $\Lambda_{\mathbf t}=\sum_{i=0}^n|l_i|$, and $\lambda_i(\mathbf t)=\max_{t_{i-1}\le x\le t_i}\Lambda_{\mathbf t}(x)$ for $i\in[1,n]$. Bernstein (1931) conjectured that $\|P_{\mathbf t}\|$ is smallest at a node vector whose Lebesgue function equioscillates, meaning it has the same maximum on every subinterval, $\lambda_1(\mathbf t)=\cdots=\lambda_n(\mathbf t)$. Erdős (the paper's [7], Acta Math. Acad. Sci. Hungar. 9 (1958)) conjectured further that exactly one $\mathbf t$ makes $\Lambda_{\mathbf t}$ equioscillate and that $\min_i\lambda_i(\mathbf t)\le\lambda^*:=\inf_{\mathbf s\in T}\|P_{\mathbf s}\|$ for every $\mathbf t\in T$ (the paper's (1)); the paper traces (1) back to [6], Bull. Amer. Math. Soc. 53 (1947), in the form "$\min_i\lambda_i(\mathbf t)$ achieves its maximum when $\Lambda_{\mathbf t}$ equioscillates" (p. 290).
• Kilgore's theorem (p. 291, from the paper's [8]): an optimal node vector, one with $\|\Lambda_{\mathbf t}\|=\lambda^*$, always has an equioscillating Lebesgue function. Section 2 outlines its proof.
• Theorem 1 (p. 295): the map $\Gamma$ that sends $\mathbf t$ to the successive differences $(\lambda_{i+1}(\mathbf t)-\lambda_i(\mathbf t))_{i=1}^{n-1}$ of its subinterval maxima carries $T$ homeomorphically onto all of $\mathbb R^{n-1}$. In particular exactly one $\mathbf t$ has $\Gamma(\mathbf t)=0$, and with Kilgore's theorem this proves Bernstein's conjecture. Corollary (p. 295): the interpolation norm at the equioscillating node vector is strictly smaller than at any other node vector in $T$. The proof (pp. 295--298) uses Lemma 3 ($\Gamma$ is a local homeomorphism), Lemma 4 ($\Gamma$ sends the boundary of $T$ to infinity) and a covering-space theorem (Theorem A).
• Theorem 2 (p. 298): no node vector is dominated subinterval by subinterval by a different one: $\lambda_i(\mathbf s)\le\lambda_i(\mathbf t)$ for all $i=1,\dots,n$ forces $\mathbf s=\mathbf t$. Hence $\lambda^*\in[\min_i\lambda_i(\mathbf t),\max_i\lambda_i(\mathbf t)]$ for every $\mathbf t\in T$, which is Erdős's conjecture (1). The proof lifts a curve through the local homeomorphism $\mathbf t\mapsto(\lambda_i(\mathbf t))_{i=2}^n$ of $T$ into $\mathbb R^{n-1}$.
• Theorem 3 (p. 301): for trigonometric interpolation on $[0,2\pi]$ at $2n+1$ nodes, $\|P_{\mathbf t}\|=\lambda^*$ exactly when the nodes are equidistant, $\mathbf t^*=(2\pi i/(2n+1))_{i=1}^{2n}$ (the print has $(i/(2n+1))_1^{2n}$, without the factor $2\pi$ its period requires), in which case $\Lambda_{\mathbf t}$ equioscillates; moreover $\min_i\lambda_i(\mathbf t)<\lambda^*<\max_i\lambda_i(\mathbf t)$ for every $\mathbf t\ne\mathbf t^*$.
• Added in proof (p. 303): after completion of the work in March 1977, Kilgore also proved Bernstein's conjecture, by a different argument (the paper's [14]).

Compiled scope

The conjectures, Kilgore's theorem and Theorems 1--3 were checked on the page images. The proofs of Sections 2--4, including Lemmas 1--6 and the determinant identities they rest on, were read for their structure only. Nothing here is independently reviewed.

Bears on. #1129, as the proof that the nodes minimizing $\Lambda(x_1,\dots,x_n)$ are the unique system whose Lebesgue function equioscillates (Theorem 1 with Kilgore's theorem and the Corollary on p. 295); the paper takes the endpoints of the interval as nodes. #1130, as the proof of Erdős's conjecture that the least local maximum of the Lebesgue function never exceeds the optimal Lebesgue constant (Theorem 2), so in the paper's node convention $\Upsilon$ is maximized exactly by the optimal nodes and its maximum is $\lambda^*$; the paper does not estimate $\lambda^*$ in terms of $n$, so the $\log n$ question is not addressed here.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.