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 In mathematical analysis, a modulus of continuity is a function ω : , ∞→ , ∞used to measure quantitatively the uniform continuity of functions. So, a function ''f'' : ''I'' → R admits ω as a modulus of continuity if and only if :, f(x)-f ...

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 Dunham Jackson (July 24, 1888 in Bridgewater, Massachusetts – November 6, 1946) was a mathematician who worked within approximation theory, notably with trigonometrical and orthogonal polynomials. He is known for Jackson's inequality. He ...

: :Theorem 1: If

to \C

is an

r

times differentiable

periodic function A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to des ...

such that ::

\left , f^(x) \right ,  \leq 1, \qquad x\in,2\pi

:then, for every positive integer

n

, there exists a

trigonometric polynomial In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(''nx'') and cos(''nx'') with ''n'' taking on the values of one or more natural numbers. The c ...

T_

of degree at most

n-1

such that ::

\left , f(x) - T_(x) \right ,  \leq \frac, \qquad x\in,2\pi

: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): :

C(r) = \frac \sum_^\infty \frac~.

Jackson also proved the following generalisation of Theorem 1: :Theorem 2: One can find a

T_n

of degree

\le n

such that ::

, f(x) - T_n(x),  \leq \frac, \qquad x\in,2\pi

:where

\omega(\delta, g)

denotes the

of function

g

with the step

\delta.

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

2\pi

-periodic continuous function, there exists a

T_n

of degree

\le n

such that ::

, f(x)-T_n(x), \leq c(k)\omega_k\left(\tfrac,f\right),\qquad x\in,2\pi

:where constant

c(k)

depends on

k\in\N,

and

\omega_k

is the

k

-th order modulus of smoothness. For

k=1

this result was proved by Dunham Jackson.

Antoni Zygmund Antoni Zygmund (December 25, 1900 – May 30, 1992) was a Polish mathematician. He worked mostly in the area of mathematical analysis, including especially harmonic analysis, and he is considered one of the greatest analysts of the 20th century. ...

proved the inequality in the case when

k=2, \omega_2(t,f)\le ct, t>0

in 1945.

Naum Akhiezer Naum Ilyich Akhiezer ( uk, Нау́м Іллі́ч Ахіє́зер; russian: link=no, Нау́м Ильи́ч Ахие́зер; 6 March 1901 – 3 June 1980) was a Soviet and Ukrainian mathematician of Jewish origin, known for his works in appr ...

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 In mathematical analysis, constructive function theory is a field which studies the connection between the smoothness of a function and its degree of approximation. It is closely related to approximation theory. The term was coined by Sergei Berns ...

References

External links

* * Approximation theory Inequalities Theorems in approximation theory {{mathanalysis-stub