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Statement

For nodes $x_0,\dots,x_{n-1}\in[-1,1]$, distinct as the abstract says, $l_i$ is the Lagrange cardinal polynomial attached to $x_i$: the polynomial of degree at most $n-1$ with $l_i(x_i)=1$ and $l_i(x_j)=0$ for $j\ne i$. Put

$$M_n=\min_{x_0,\dots,x_{n-1}\in[-1,1]}\int_{-1}^1\sum_{i=0}^{n-1}l_i(x)^2\,dx .$$

Theorem 1 (p. 1). There are constants $\varepsilon>0$ and $n_0$ such that

$$M_n\le2-\frac{1+\varepsilon}{n}\qquad\text{for every }n\ge n_0 .$$

Consequently $M_n\ne2-(1+o(1))\frac1n$: the quantity $n(2-M_n)$, at least $1+\varepsilon$ for every $n\ge n_0$, does not tend to $1$.

Source. Perturbed Lobatto nodes, preprint (2026), 11 pp., posted by Liam Price; Theorem 1 on p. 1, the node family on pp. 1--2, (28) and (29) on p. 6, (48)--(54) on p. 8, (55) on p. 9, (64)--(66) on p. 10 and the completion on pp. 10--11. The edition read, the Overleaf compile of 2026-10-14, is identified on the source card.

Read depth. Claims checked: Theorem 1 and the displays stated below were read clause by clause on the page images. The proof was read for structure only. Of it, only the arithmetic of (55), of (64) from (61) and (63), and of (66) from (65) was redone by hand; the rank-three formula (17) and the uniform bound (53), on which the theorem rests, were not checked. Nothing here is independently reviewed.

The node family and the uniform expansion

The family (Section 1, pp. 1--2). $P_m$ is the Legendre polynomial normalized by $P_m(1)=1$. For $n\ge4$ put $N=n-1\ge3$ and, for real $a$,

$$Q_{N,a}(x)=\int_{-1}^x\bigl(P_N(t)+aP_{N-2}(t)\bigr)\,dt =\frac{P_{N+1}(x)-P_{N-1}(x)}{2N+1}+a\,\frac{P_{N-1}(x)-P_{N-3}(x)}{2N-3} =(x^2-1)\Bigl[\frac{P_N'(x)}{N(N+1)}+a\,\frac{P_{N-2}'(x)}{(N-1)(N-2)}\Bigr].$$

The nodes are the $N+1=n$ roots of $Q_{N,a}$. The paper shows, for every $N\ge3$ and $|a|\le a_*=1/100$, that these roots are real and simple,

$$-1=x_0(a)<x_1(a)<\dots<x_{N-1}(a)<x_N(a)=1,$$

and real-analytic in $a$. At $a=0$, $Q_{N,0}(x)=(x^2-1)P_N'(x)/(N(N+1))$ and the nodes are the Legendre--Lobatto nodes $\xi_0=-1<\xi_1<\dots<\xi_N=1$, the endpoints and the zeros of $P_N'$ (p. 7).

The uniform bound (Section 3, pp. 6--8). Write $I_N(a)=\int_{-1}^1\sum_{i=0}^Nl_i(x)^2\,dx$ for the nodes $x_i(a)$ (p. 3). The paper takes

$$a_0=\min\Bigl\{\frac1{100},\ \frac1{4C_{\mathcal A}\sup_N\lvert\mathcal A(0)^{-1}\rvert}\Bigr\}$$

((50), p. 8), where $\mathcal A(a)$ is the $3\times3$ matrix of the formula $I_N(a)=2-\operatorname{tr}(\mathcal A(a)^{-1}C(a)B(a))$ ((28), p. 6), $\mathcal A(0)^{-1}$ is bounded uniformly in $N$ ((48), p. 8), and $C_{\mathcal A}$ is an absolute constant with $|\mathcal A'(a)|+|\mathcal A''(a)|\le C_{\mathcal A}$ on $|a|\le a_*$ ((49), p. 8). It proves

$$\lvert I_N''(a)\rvert\le\frac CN\qquad\text{for }\lvert a\rvert\le a_0$$

((29), p. 6, proved as (53), p. 8), with $C$ and $a_0>0$ absolute constants, independent of $N$ and $a$. None of $C$, $C_{\mathcal A}$ and $a_0$ is evaluated.

The expansion (Sections 4--5, pp. 8--10). Taylor's theorem with that bound gives $I_N(a)=I_N(0)+aI_N'(0)+O(a^2/N)$ uniformly for $|a|\le a_0$ ((54), p. 8). At the Lobatto nodes $I_N(0)=2-2/(2N+1)$ ((55), p. 9), and the first variation is $I_N'(0)=-4(N+1)/(3(N-1)(2N+1))$ ((64), p. 10). Hence, uniformly for $|a|\le a_0$, the paper's (65) (p. 10):

$$I_N(a)=2-\frac2{2N+1}-a\,\frac{4(N+1)}{3(N-1)(2N+1)}+O\Bigl(\frac{a^2}N\Bigr).$$

In terms of $n=N+1$, the display before (66) and (66) itself (p. 10) read

$$n\bigl(2-I_N(a)\bigr)=\frac{2n}{2n-1}+a\,\frac{4n^2}{3(n-2)(2n-1)}+O(a^2) =1+\frac{2a}3+O(a^2)+o(1),$$

the $o(1)$ tending to $0$ as $n\to\infty$. With $C_1$ the implicit constant of $O(a^2)$ in (66), the paper fixes $a\in(0,a_0)$ with $2a/3-C_1a^2>0$ and concludes that $n(2-I_N(a))\ge1+\varepsilon$ for some $\varepsilon>0$ and all large $n$ (pp. 10--11). The $n$ roots of $Q_{n-1,a}$ are admissible nodes, so $M_n\le I_{n-1}(a)\le2-(1+\varepsilon)/n$.

Read off the printed constants, and not stated in the paper: the $\varepsilon$ this argument yields is at most $2a/3$, hence less than $1/150$, since $a<a_0\le1/100$.

Proof pointer

Sections 1--5, pp. 1--11, in outline.

• Roots (pp. 1--2). The endpoints $\pm1$ are roots, with $Q_{N,a}'(\pm1)=(\pm1)^N(1+a)\ne0$. The interior roots are the zeros of $P_N'+\gamma_NaP_{N-2}'$, $\gamma_N=N(N+1)/((N-1)(N-2))$, a positive multiple of the characteristic polynomial of a Jacobi matrix with zero diagonal and positive off-diagonal entries built from the Gegenbauer recurrence ((2), p. 2), so they are real and simple; interlacing and the signs at $\pm1$ put them in $(-1,1)$.
• Kernel reduction (pp. 2--6). With $V_{im}=\phi_m(x_i)$ for the orthonormal Legendre polynomials $\phi_m$, the integral is $\operatorname{tr}((VV^T)^{-1})=\operatorname{tr}K(a)^{-1}$, where $K(a)_{ij}=K_N(x_i,x_j)$ is the Christoffel kernel at the nodes ((3)). Identities at the roots ((5)--(8)) and the Christoffel--Darboux formula give $K(a)=D(a)+U(a)C(a)U(a)^T$ with $D(a)$ positive diagonal, $U$ of size $(N+1)\times3$ and $C(a)$ of size $3\times3$ ((17)). The reciprocals of the diagonal entries are the interpolatory quadrature weights, which sum to $2$ ((26)--(27)), and the Woodbury identity gives (28).
• Uniform bound (pp. 6--8). The node velocities satisfy $|x_i'|,|x_i''|\le C(1-x_i^2)/N$ ((9)--(10), p. 3). Bounds on the weights along the moving nodes ((30)--(40)) give $|M|+|M'|+|M''|\le C/N$ and $|B|+|B'|+|B''|\le C/N^2$ for $M=U^T\Lambda U$ and $B=U^T\Lambda^2U$ ((44), (46)), with $|C|+|C'|+|C''|\le CN$ ((18)). A perturbation of the explicit $\mathcal A(0)=I_3+C(0)M(0)$ ((48)) bounds $\mathcal A(a)^{-1}$ and its first two derivatives for $|a|\le a_0$ ((51)--(52)), which gives (53).
• Value and first variation at $a=0$ (pp. 8--10). There $K(0)$ is diagonal plus rank one, and the Sherman--Morrison formula gives (55). Differentiating $l_k(x_j)=\delta_{kj}$ along the motion of the nodes, with the node velocities (60) and the Lobatto identities $\sum_i\xi_i=0$ and $\sum_i\xi_i^2=N(N+1)/(2N-1)$ ((47)), gives (64).
• Conclusion (pp. 10--11). (54), (55) and (64) give (65) and (66), and a fixed small $a>0$ gives the theorem.

Dependencies

The write-up labels no other result and has no bibliography. It uses without proof the Legendre recurrence, differential equation and integration identity, the Christoffel--Darboux formula, the Gegenbauer recurrence, the reality, simplicity and interlacing of the zeros of Jacobi-matrix characteristic polynomials, the Woodbury and Sherman--Morrison formulas, and the exactness of the Lobatto quadrature for $1$ and $x^2$.

Bears on

• Problem 1131: the theorem gives $n(2-\min I)\ge1+\varepsilon$ for every $n\ge n_0$, so the displayed asymptotic $\min I=2-(1+o(1))/n$ fails and the second question is answered no. It proves no lower bound for $\min I$, so the first question stays open. At $a=0$ the nodes are the roots of the integral of the Legendre polynomial $P_{n-1}$, and (55) is the value $2-2/(2n-1)$ that the problem page records as the upper bound of Erdős, Szabados, Varma and Vértesi. The theorem is the result behind the pending claim of Price.