Rack 2015 optimal cubic lagrange interpolation extremal node

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lemma_4_3: Rack and Vajda's two descriptions of the constant b that bounds the parameters of the optimal four-node systems: the unique positive root of an explicit integer polynomial of degree 18 (Lemma 4.3), and an explicit expression by radicals in terms of t (Lemma 4.4).

proposition_2_8: The known result, which Rack and Vajda record with its proofs in their references [8] (de Boor and Pinkus) and [14] (Kilgore), that a canonical node system on [-1,1] whose Lebesgue function has equal local maxima is optimal.

theorem_2_5: Rack and Vajda's strengthening of the known non-uniqueness of optimal interpolation nodes: for each n >= 3 there are uncountably many node systems of n points in [-1,1] attaining the minimal Lebesgue constant.

theorem_4_2: Rack and Vajda's description of the zero-symmetric four-node systems on [-1,1] that minimize the Lebesgue constant of cubic Lagrange interpolation: they are the scaled canonical systems -1/beta < -t/beta < t/beta < 1/beta for beta in [1,b], with beta = b giving the shortest one (Example 4.5).

theorem_5_2: Rack and Vajda's complete description of the four-node systems in [-1,1] that minimize the Lebesgue constant of cubic Lagrange interpolation: they are exactly the systems (5.1), the images of the optimal canonical system -1 < -t < t < 1 under the affine map of [alpha, beta] onto [-1,1], for arbitrary alpha in [-b,-1] and beta in [1,b].

theorem_5_4: Rack and Vajda's second description of all optimal four-node systems on [-1,1]: the outer nodes range over the region (5.4) or (5.5), bounded with the constant b, and the inner nodes are then the fixed affine combinations (5.6) and (5.7) of the outer ones with the constant t; Theorem 5.5 gives the ranges (5.8) and (5.9) of the inner nodes.

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Rack, Heinz-Joachim and Vajda, Robert, Optimal cubic Lagrange interpolation: Extremal node systems with minimal Lebesgue constant. Stud. Univ. Babeş-Bolyai Math. 60 (2015), no. 2, 151--171. The article prints no license; the journal's open access policy page states "All articles published in Studia UBB Mathematica are fully open access: immediately freely available to read, download and share." and that "Authors can re/use the material however they want as long as it fits the NC ND terms of the license Creative Commons." (https://www.cs.ubbcluj.ro/journal/studia-mathematica/journal/oap, read 2026-10-02): a Creative Commons Attribution-NonCommercial-NoDerivatives license, named by its NC ND terms with no version stated, by the journal's policy rather than an article-level statement.

Rack and Vajda determine all optimal node systems for cubic Lagrange interpolation on [-1,1] (n = 4 nodes, degree n-1 = 3): every four-node system x1* < x2* < x3* < x4* in [-1,1] attaining the minimal Lebesgue constant is described explicitly, in two equivalent forms built from two constants given by radicals. The constant t = 0.4177913013... is the inner node of the optimal canonical system -1 < -t < t < 1 (3.7), which the paper records as solved in its references [23], [24] together with the minimal Lebesgue constant of value 1.4229195732... (pp. 155-156). The constant b = 1.0433133411... is the point beyond 1 where the Lebesgue function of that canonical system reaches the minimal Lebesgue constant (Lemmas 4.3 and 4.4, pp. 157-158). Theorem 5.2 (p. 160) shows that the optimal systems are exactly the affine images of the canonical system under the maps of [alpha, beta] onto [-1,1] with alpha in [-b,-1] and beta in [1,b]; Theorem 5.4 (p. 161) describes the same systems by the admissible region of the two outer nodes, which then fix the inner ones, and Theorem 5.5 gives the ranges of the inner nodes. Theorem 4.2 (p. 157) is the zero-symmetric case (-1/beta, -t/beta, t/beta, 1/beta), beta in [1, b], with beta = 1 the canonical system and beta = b the shortest-interval system of Example 4.5. Lemma 4.3 characterizes b as the unique positive root of an explicit integer polynomial of degree 18. For general n, Theorem 2.5 (p. 154) amplifies the non-uniqueness result of the paper's reference [17] (Theorem 2): for each n >= 3 there are uncountably many optimal node systems in [-1,1]. Proposition 2.8 (p. 155) records the known result, proved in the paper's references [8] and [14], that a canonical node system whose Lebesgue function equioscillates is extremal. The proofs (Section 6, pp. 163-168) use symbolic computation in Mathematica at key steps: RootReduce for Lemma 4.3, and quantifier elimination through Resolve for the parameter ranges in Theorem 5.2 and for Theorem 5.4. The paper states that before it the optimal canonical system and the minimal Lebesgue constant were known explicitly ([23], [24]) and the zero-symmetric systems implicitly (Tureckii [29], [30]), and that optimal systems that are not zero-symmetric are not mentioned there.

Source: https://www.cs.ubbcluj.ro/journal/studia-mathematica/archive/2015-2/Cuprins2015_2.htm.

Bears on. #1129: for n = 4 nodes in [-1,1], Theorem 5.2 and Theorem 5.4 describe every node system minimizing the Lebesgue constant, the problem's quantity; this description covers n = 4 only. Theorem 2.5 shows that for every n >= 3 the minimizing node systems are not unique.

Read status. Claims checked: Theorems 2.5, 4.2, 5.2, 5.4 and 5.5, Lemmas 4.3 and 4.4, Proposition 2.8 and the cubic constants (3.1)-(3.9) were read clause by clause on the page images of the print, and the proofs in Section 6 were followed in outline. The computer-algebra steps were not re-run, and the results the paper cites from its references were not read. Nothing here is independently reviewed.

Results.

• Theorem 5.2 (p. 160): the optimal four-node systems on [-1,1] are exactly the systems (5.1), the affine images of -1 < -t < t < 1 for parameters alpha in [-b,-1] and beta in [1,b].
• Theorems 5.4 and 5.5 (p. 161): the same systems described by the region (5.4)-(5.5) of the outer nodes, with the inner nodes given by (5.6)-(5.7), and the ranges (5.8)-(5.9) of the inner nodes.
• Theorem 4.2 (p. 157): the optimal zero-symmetric four-node systems are (-1/beta, -t/beta, t/beta, 1/beta) with beta in [1, b]; the page also states Example 4.5 (pp. 158-159).
• Lemmas 4.3 and 4.4 (pp. 157-158): b is the unique positive root of an explicit integer polynomial of degree 18, and b is given by radicals in terms of t.
• Theorem 2.5 (p. 154): for each n >= 3 there are uncountably many optimal node systems in [-1,1].
• Proposition 2.8 (p. 155): a canonical node system whose Lebesgue function equioscillates is extremal; recorded from the paper's references [8] and [14], not proved here.

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