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Lipschitz integral condition

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A restriction on the behaviour of increase of a function in an integral metric. A function in a space with satisfies the Lipschitz integral condition of order on with constant if

(*)

for all . In this case one writes , or , . For the case of a periodic function (with period ) the Lipschitz integral condition is defined similarly, only in inequality (*) the upper limit of integration must be replaced by .

References

[1] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)
[2] S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian)
[3] O.V. Besov, V.P. Il'in, S.M. Nikol'skii, "Integral representations of functions and imbedding theorems" , Wiley (1978) (Translated from Russian)
How to Cite This Entry:
Lipschitz integral condition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lipschitz_integral_condition&oldid=47671
This article was adapted from an original article by A.V. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article