Difference between revisions of "Dini-Lipschitz criterion"
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If a continuous -periodic function
satisfies the condition
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where is the modulus of continuity of the function
, then its Fourier series converges uniformly to it on the entire real axis. The criterion was demonstrated by U. Dini [1], and also by R. Lipschitz for the special case when
,
, for any
[2]. The Dini–Lipschitz criterion is a final (sharp) statement in the following sense. If
is an arbitrary modulus of continuity satisfying the condition
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then there exists a continuous -periodic function
whose Fourier series diverges at some point, while
satisfies the condition
.
References
[1] | U. Dini, "Sopra la serie di Fourier" , Pisa (1872) |
[2] | R. Lipschitz, "De explicatione per series trigonometricas instituenda functionum unius variabilis arbitrariarum, etc." J. Reine Angew. Math. , 63 : 2 (1864) pp. 296–308 |
[3] | H. Lebesgue, "Sur la répresentation trigonométrique approchée des fonctions satisfiasants à une condition de Lipschitz" Bull. Soc. Math. France , 38 (1910) pp. 184–210 |
[4] | S.M. Nikol'skii, "On the Dini–Lipschitz condition for convergence of Fourier series" Doklady Akad. Nauk SSSR , 73 : 3 (1950) pp. 457–460 (In Russian) |
[5] | N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian) |
How to Cite This Entry:
Dini-Lipschitz criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dini-Lipschitz_criterion&oldid=22357
Dini-Lipschitz criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dini-Lipschitz_criterion&oldid=22357
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article