Sharp integral bounds for trigonometric Lebesgue-type sums
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For a real trigonometric polynomial P of degree n with 2n distinct zeros xₖ in one period, how small can the integral of the sum ∑ₖ|P(x)|/|P′(xₖ)| be? The sharp proposed lower value is 8.
Why it matters and what counts as progress
Why it matters: The integral bound controls an interpolation quantity differently from its known sup-norm bound.
What counts as progress: A universal integral lower bound of 8 with a matching sinusoid, or a polynomial giving a smaller integral under the same zero assumptions.
How the problem developed
- 23 Mar 2026 · Lower bound [1]
Terence Tao, with AI-assisted proof development Proved that for a real trigonometric polynomial of degree n with 2n distinct roots in one period, the sum of |P(x)|/|P′(xₖ)| has supremum at least 2 and integral at least 8; a sinusoid attains both bounds.
AI contribution
Local Bernstein theory, and lower bounds for Lebesgue constants
For a real degree-n trigonometric polynomial with 2n distinct zeros on a period, Tao’s Theorem 1.13(ii) gives an integral of its Lebesgue-type sum at least 8, with equality for a sinusoid. The bound follows by multiplying two factors in Lemma 1.14.
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