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.
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Statement: trigonometric polynomials
For trigonometric polynomials, the following was proved by Dunham Jackson:
Theorem 1: If
then, for every positive integer
where
The Akhiezer–Krein–Favard theorem gives the sharp value of
Jackson also proved the following generalisation of Theorem 1:
Theorem 2: Denote by
An even more general result of four authors can be formulated as the following Jackson theorem.
Theorem 3: For every natural number
where constant
For
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.