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)