Bernstein's quarter-logarithm local bound

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Source. Bernstein 1931, section 4, especially equations (27)--(34), printed pp. 1036--1040 / PDF pp. 12--16, in the complete source. The proof below combines the source's midpoint and telescoping method with the separately attributed local gap companion.

Let $I=[\alpha,\beta]\subseteq[-1,1]$ have fixed length $L>0$. For degree $d$ and any $d+1$ distinct nodes in $[-1,1]$, let $F$ be the ordinary Lebesgue function. Uniformly in those nodes,

$$\max_{x\in I}F(x)> \frac14\log d-O_I(\log\log\log d) \qquad(d\longrightarrow\infty). \tag{L}$$

More precisely, set $m=\lfloor d/2\rfloor$. Whenever $d\ge16$ and

$$8\log(2\log d)<mL, \tag{L0}$$

the following finite bound holds:

$$\max_{x\in I}F(x)> \frac14\log\frac{Lm}{8\log(2\log d)}. \tag{L1}$$

Condition (L0) holds for every sufficiently large $d$, with a threshold depending only on $L$.

Proof. Write $M=\max_I F$. If $M>\log d$, then (L1) follows immediately: $L\le2$, $m\le d/2$, and $\log(2\log d)>1$ imply that the argument of its logarithm is less than $d$, so its right side is less than $\tfrac14\log d<\log d$.

It remains to treat $M\le\log d$. The gap companion gives a bound for every node-free subinterval of $I$:

$$D=\frac{2\log(2M)}m \le D_0=\frac{2\log(2\log d)}m<\frac L4. \tag{L2}$$

Choose $\xi\in I$ with $|A(\xi)|=\max_I|A|$. It is not a node. Suppose first that $\xi\ge(\alpha+\beta)/2$. Let $u$ be the leftmost node in $I$; the gap bound gives $u<\alpha+D$. Consequently,

$$\xi-u>\frac L2-D>\frac L4>D.$$

Let $v$ be the last node in $I$ strictly to the left of $\xi$. Such a node exists because $u<\xi$. The node-free interval $(v,\xi)$ has length less than $D$. Also $v\ne u$, since otherwise $\xi-u>D$ would be a node-free interval in $I$.

All consecutive-node midpoints between $u$ and $v$ lie in $I$ and have nodal-polynomial modulus at most $|A(\xi)|$. The one-sided telescoping inequality therefore gives

$$M\ge F(\xi)> \frac14\log\frac{\xi-u}{\xi-v} >\frac14\log\frac{L}{4D} \ge\frac14\log\frac{L}{4D_0},$$

which is (L1). If $\xi\le(\alpha+\beta)/2$, use the rightmost node of $I$ and the first node to the right of $\xi$ instead. The same proof, with the line reversed, gives the identical bound. This includes a nodal-polynomial maximum at either endpoint of $I$.

Finally, $m\ge d/3$ for $d\ge2$, so (L1) implies

$$M> \frac14\log d -\frac14\log\log(2\log d) +\frac14\log\frac L{24}.$$

Since $\log\log(2\log d)=\log\log\log d+O(1)$ for sufficiently large $d$, this proves (L), and in fact the displayed deduction gives $\tfrac14\log d-\tfrac14\log\log\log d-O_I(1)$.

What is and is not attributed to the source. Equation (34) prints the coefficient $1/4$ and the triple-logarithmic error at a selected point of the nodal-polynomial argument. The finite local gap reduction and the explicit case $M>\log d$ above belong to the compilation companion. In that large-$M$ case, the conclusion concerns a maximizing point of $F$; this proof does not assert that an arbitrary selected maximizer of $|A|$ also has that value.

Endpoints, uniformity, and equality. Distinct nodes are essential. The interval has positive length and is fixed independently of $d$; it may touch $-1$ or $1$. The threshold in (L0) depends only on its length, not on the nodes. The finite bound is strict, but it is not an extremizer classification or an assertion of optimal constants.

Relation to the problems. In Problem 1153, the node count is $N=d+1$ and its $\lambda$ is this $F$. Because $\log(N-1)=\log N+O(1/N)$, (L) yields the same historical $1/4$ coefficient in that convention. It does not give E1153's $2/\pi$ coefficient. For Problem 1129, taking $I=[-1,1]$ yields only a weak lower bound for the global minimum, not its minimizing configurations. The selected point may change with $d$, so this argument supplies no fixed-point or almost-everywhere conclusion for Problem 1132.

Dependencies. The complete chain is the interpolation extremum, equation (27), equations (29)--(31), equation (32), and the local gap companion. No external theorem proof is imported.

Proof scope. Complete rewritten local maximum argument with an separately attributed elementary companion; the selected chain and companion were independently reviewed on 6 September 2026 (components C6 and C5 of the local-chain review). No problem-status, publication-acceptance, or formal-verification credit follows.