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''Jackson operator''
 
''Jackson operator''
  
 
An integral of the form
 
An integral of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054010/j0540101.png" /></td> </tr></table>
+
$$
 +
U _ {n} ( f , x )  =
 +
\frac{1} \pi
 +
\int\limits _ {- \pi } ^ { {+ }  \pi } f
 +
( x + u ) K _ {n} ( u)  du ,
 +
$$
  
 
in which the expression
 
in which the expression
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054010/j0540102.png" /></td> </tr></table>
+
$$
 +
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054010/j0540103.png" /> in the modulus of continuity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054010/j0540104.png" /> or in the modulus of continuity of its derivative of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054010/j0540105.png" />. Jackson's singular integral is a positive operator and is a trigonometric polynomial of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054010/j0540106.png" />; its kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054010/j0540107.png" /> can be represented in the form
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054010/j0540108.png" /></td> </tr></table>
+
$$
 +
K _ {n} ( u)  = A + \rho _ {1} ^ {2n - 2 } \cos  t + \dots +
 +
\rho _ {2n - 2 }  ^ {2n - 2 } \cos  ( 2n - 2 ) t ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054010/j0540109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054010/j05401010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054010/j05401011.png" />. The estimate
+
where $  A = 1 / 2 $
 +
and $  \rho _ {1} ^ {2n - 2 } = 1 - 3 / ( 2n  ^ {2} ) $,  
 +
$  n = 1 , 2 , . . . $.  
 +
The estimate
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054010/j05401012.png" /></td> </tr></table>
+
$$
 +
| U _ {n} ( f , x ) - f( x) |  \leq  6 \omega \left ( f ,
 +
\frac{1}{n}
 +
\right )
 +
$$
  
 
is valid.
 
is valid.

Latest revision as of 22:14, 5 June 2020


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