Jackson's inequality

Statement: trigonometric polynomials

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

Theorem 1: If f : [ 0 , 2 π ] → C is an r times differentiable periodic function such that

| f ( r ) ( x ) | ≤ 1 , 0 ≤ x ≤ 2 π ,

then, for every positive integer n , there exists a trigonometric polynomial T n − 1 of degree at most n − 1 such that

| f ( x ) − T n − 1 ( x ) | ≤ C ( r ) n r , 0 ≤ x ≤ 2 π ,

where C ( r ) depends only on r .

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:

Theorem 2: Denote by ω ( δ , f ( r ) ) the modulus of continuity of the r -th derivative of f with the step δ . Then one can find a trigonometric polynomial T n of degree ≤ n such that

| f ( x ) − T n ( x ) | ≤ C 1 ( r ) ω ( 1 / n , f ( r ) ) n r , 0 ≤ x ≤ 2 π .

An even more general result of four authors can be formulated as the following Jackson theorem.

Theorem 3: For every natural number n , if f is 2 π -periodic continuous function, there exists a trigonometric polynomial T n of degree ≤ n such that

| f ( x ) − T n ( x ) | ≤ c ( k ) ω k ( 1 n , f ) , x ∈ [ 0 , 2 π ] ,

where constant c ( k ) depends on k ∈ N , and ω k is the k -th order modulus of smoothness.

For k = 1 this result was proved by Dunham Jackson. Antoni Zygmund proved the inequality in the case when k = 2 , ω 2 ( t , f ) ≤ c t , t > 0 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.