Jackson's inequality

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.

For trigonometric polynomials, the following was proved by Dunham Jackson:

Theorem 1: If ${\displaystyle f:[0,2\pi ]\to \mathbb {C} }$ is an ${\displaystyle r}$ times differentiable periodic function such that

${\displaystyle |f^{(r)}(x)|\leq 1,\quad 0\leq x\leq 2\pi ,}$

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