Statement: trigonometric polynomials
For trigonometric polynomials, the following was proved by Dunham Jackson:
- Theorem 1: If
is an
times differentiable periodic function such that 
- then, for every positive integer
, there exists a trigonometric polynomial
of degree at most
such that 
- where
depends only on
.
The Akhiezer – Krein – Favard theorem gives the sharp value of
(called the Akhiezer–Krein–Favard constant):

Jackson also proved the following generalisation of Theorem 1:
- Theorem 2: One can find a trigonometric polynomial
of degree
such that 
- where
denotes the modulus of continuity of function
with the step 
An even more general result of four authors can be formulated as the following Jackson theorem.
- Theorem 3: For every natural number
, if
is
-periodic continuous function, there exists a trigonometric polynomial
of degree
such that 
- where constant
depends on
and
is the
-th order modulus of smoothness.
For
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
in 1956. For
this result was established by Sergey Stechkin in 1967.
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