The pointwise interpolation extremum

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Source. Bernstein 1931, equation (1), printed p. 1025 / PDF p. 1, and its explanation on printed p. 1026 / PDF p. 2, in the complete source. The argument below uses degree at most $d$ and $d+1$ nodes; Bernstein calls the degree $n$.

Let $d\ge0$, let $a_0<\cdots<a_d$ be real, and put

$$A(x)=\prod_{j=0}^d(x-a_j),\qquad \ell_j(x)=\prod_{i\ne j}\frac{x-a_i}{a_j-a_i},\qquad F(x)=\sum_{j=0}^d|\ell_j(x)|.$$

These are polynomials, including at their nodes. Away from the nodes, $\ell_j(x)=A(x)/((x-a_j)A'(a_j))$. For every real $x$,

$$F(x)= \max\left\{|P(x)|:\deg P\le d,\ |P(a_j)|\le1\ (0\le j\le d)\right\}. \tag{E}$$

The maximum is the same for real or complex coefficients. In particular, $F(x)\ge1$, $F(a_j)=1$, and $F$ is continuous.

Proof. The polynomial $P-\sum_jP(a_j)\ell_j$ has degree at most $d$ and vanishes at $d+1$ distinct points, so it is zero. Thus

$$|P(x)|\le\sum_j|P(a_j)|\,|\ell_j(x)|\le F(x).$$

For fixed real $x$, assign $P(a_j)=\operatorname{sgn}\ell_j(x)$ when $\ell_j(x)\ne0$, and assign any value of modulus at most one to the remaining node data. The interpolating polynomial has real coefficients and value $F(x)$ at $x$. This proves attainment and (E). Interpolating the constant polynomial gives $\sum_j\ell_j=1$, hence $F\ge1$; evaluation at a node and continuity give the other assertions.

Equality and endpoints. At a non-node $x$, every $\ell_j(x)$ is nonzero. Equality $|P(x)|=F(x)$ holds precisely when all node values have modulus one and the numbers $P(a_j)\ell_j(x)$ have one common complex argument. For real coefficients this allows a common sign. At $x=a_j$, equality requires only $|P(a_j)|=1$. Also, $F(x)=1$ precisely when all the real numbers $\ell_j(x)$ are nonnegative. All statements apply at $\pm1$ when the nodes lie in $[-1,1]$; the rational expression is never evaluated through a zero denominator.

Dependencies. Polynomial interpolation and the triangle inequality, both proved or applied explicitly above.

Proof scope. Complete rewritten elementary argument; independently reviewed on 6 September 2026 (component C1 of the local-chain review). No formal verification is claimed.

Bears on. Problem 1153; Problem 1129, common global minimax quantity.