In approximation theory, Jackson's inequality is an inequality bounding the value of function's best approximation by algebraic or trigonometric polynomials in terms of the modulus of continuity or modulus of smoothness of the function or of its derivatives. Informally speaking, the smoother the function is, the better it can be approximated by polynomials.

Statement: trigonometric polynomials

For trigonometric polynomials, the following was proved by Dunham Jackson: The Akhiezer–Krein–Favard theorem gives the sharp value of C(r) (called the Akhiezer–Krein–Favard constant): Jackson also proved the following generalisation of Theorem 1: An even more general result of four authors can be formulated as the following Jackson theorem. For k=1 this result was proved by Dunham Jackson. Antoni Zygmund proved the inequality in the case when in 1945. Naum Akhiezer proved the theorem in the case k=2 in 1956. For k>2 this result was established by Sergey Stechkin in 1967.

Further remarks

Generalisations and extensions are called Jackson-type theorems. A converse to Jackson's inequality is given by Bernstein's theorem. See also constructive function theory.