Jackson singular integral

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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.


[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:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098