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Claim. Write $\lambda^*_4$ for the least Lebesgue constant of four nodes in $[-1,1]$, and $-1,-t,t,1$ for the optimal canonical four-node system, with $t=0.4177913013\ldots$ and $\lambda^*_4=1.4229195732\ldots$, both given by radicals. Rack and Vajda prove the following.

• Theorem 5.2, with its alternative form Theorem 5.4: the four-node systems in $[-1,1]$ that minimize the Lebesgue constant are exactly the affine images (5.1) of $-1,-t,t,1$ under the map of $[\alpha,\beta]$ onto $[-1,1]$, with $\alpha\in[-b,-1]$ and $\beta\in[1,b]$. Here $b=1.0433133411\ldots$ is the unique positive root of an integer polynomial of degree $18$ (Lemma 4.3) and is also given by radicals (Lemma 4.4); $\pm b$ are the points beyond $\pm1$ where the Lebesgue function of the canonical system reaches $\lambda^*_4$.
• Theorem 4.2, the case $\alpha=-\beta$: the zero-symmetric optimal systems are $(-1/\beta,-t/\beta,t/\beta,1/\beta)$ for $\beta\in[1,b]$.
• Theorem 2.5: for every $n\ge3$ there are uncountably many optimal systems of $n$ nodes in $[-1,1]$. This amplifies Theorem 2 of Luttmann and Rivlin, Some numerical experiments in the theory of polynomial interpolation, IBM J. Res. Develop. 9 (1965), 187–191, as the paper cites it.

The proof of Theorem 2.5 maps the optimal canonical system affinely from any $[\alpha,\beta]\supseteq[-1,1]$ on which its Lebesgue function stays at most the canonical minimum; the proof of Theorem 5.2 rescales an arbitrary optimal system onto $[-1,1]$ and uses the uniqueness of the optimal canonical system to identify it. The explicit canonical optimum $t$ and $\lambda^*_4$, which the site credits to this paper, is recalled in its Section 3 from Rack, An example of optimal nodes for interpolation, Int. J. Math. Educ. Sci. Technol. 15 (1984), 355–357, and Rack, An example of optimal nodes for interpolation revisited, Springer Proc. Math. Stat. 41 (2013), 117–120. The source card is Rack and Vajda 2015. For four nodes the result describes every minimizing choice, as Problem 1129 asks, so the claim value is answered.

Covers. The four-node instance, with all minimizers described explicitly; and, for $n\ge3$, the non-uniqueness of free-node minimizers.

Depends on. de Boor and Pinkus 1978, whose uniqueness of the optimal canonical system the proof of Theorem 5.2 uses.

Acceptance. Refereed: Studia Universitatis Babeş-Bolyai Mathematica 60 (2015), no. 2 (June 2015), 151–171. Reviewed: the site's curator, Thomas F. Bloom, labels the problem PROVED and credits Rack and Vajda with the four-node optimum (erdosproblems.com/1129, last edited 2026-01-23). The constants $b$ and $t$ are computed with symbolic computation in Mathematica, as the paper states. No formalization declares itself a formalization of this paper.