# Jackson singular integral

Jackson operator

An integral of the form

$$U _ {n} ( f , x ) = \frac{1} \pi \int\limits _ {- \pi } ^ { {+ } \pi } f ( x + u ) K _ {n} ( u) du ,$$

in which the expression

$$K _ {n} ( u) = \frac{3}{2n ( 2n ^ {2} + 1 ) } \left ( \frac{\sin ( {n u } / 2) }{\sin ( {u } / 2) } \right ) ^ {4} ,\ n = 1 , 2 \dots$$

is known as a Jackson kernel. It was first employed by D. Jackson [1] in his estimate of the best approximation of a function $f$ in the modulus of continuity $\omega ( f , 1 / n )$ or in the modulus of continuity of its derivative of order $k \geq 1$. Jackson's singular integral is a positive operator and is a trigonometric polynomial of order $2n - 2$; its kernel $K _ {n} ( u)$ can be represented in the form

$$K _ {n} ( u) = A + \rho _ {1} ^ {2n - 2 } \cos t + \dots + \rho _ {2n - 2 } ^ {2n - 2 } \cos ( 2n - 2 ) t ,$$

where $A = 1 / 2$ and $\rho _ {1} ^ {2n - 2 } = 1 - 3 / ( 2n ^ {2} )$, $n = 1 , 2 , . . .$. The estimate

$$| U _ {n} ( f , x ) - f( x) | \leq 6 \omega \left ( f , \frac{1}{n} \right )$$

is valid.

#### References

 [1] D. Jackson, "The theory of approximation" , Amer. Math. Soc. (1930) [2] I.P. Natanson, "Constructive function theory" , 1–3 , F. Ungar (1964–1965) (Translated from Russian)
How to Cite This Entry:
Jackson singular integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jackson_singular_integral&oldid=47453
This article was adapted from an original article by A.V. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article