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''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 [[#References|[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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Jackson,   "The theory of approximation" , Amer. Math. Soc.  (1930)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.P. Natanson,   "Constructive function theory" , '''1–3''' , F. Ungar  (1964–1965)  (Translated from Russian)</TD></TR></table>

{{SourceattributionEOM|Jackson singular integral}}
[[Category:Mathematics]]