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Dini criterion

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If a -periodic function which is integrable on the segment satisfies the condition

at a point for a fixed number , , and an arbitrary , then the Fourier series of at converges to . The criterion was proved by U. Dini [1]. It is a final (sharp) statement in the following sense. If is a continuous function such that the function is not integrable in a neighbourhood of the point , it is possible to find a continuous function whose Fourier series diverges at and such that

for small .

References

[1] U. Dini, "Serie di Fourier e altre rappresentazioni analitiche delle funzioni di una variabile reale" , Pisa (1880)
[2] A. Zygmund, "Trigonometric series" , 1 , Cambridge Univ. Press (1988)
How to Cite This Entry:
Dini criterion. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Dini_criterion&oldid=16762
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article