# Jackson singular integral

From Encyclopedia of Mathematics

*Jackson operator*

An integral of the form

in which the expression

is known as a Jackson kernel. It was first employed by D. Jackson [1] in his estimate of the best approximation of a function in the modulus of continuity or in the modulus of continuity of its derivative of order . Jackson's singular integral is a positive operator and is a trigonometric polynomial of order ; its kernel can be represented in the form

where and , . The estimate

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. A.V. Efimov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Jackson_singular_integral&oldid=15968

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098