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Hardy-Littlewood criterion

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for the convergence of a Fourier series

If a -periodic function is such that

and if its Fourier coefficients satisfy the conditions

for some , then the Fourier series of at converges to .

The criterion was established by G.H. Hardy and J.E. Littlewood [1].

References

[1] G.H. Hardy, J.E. Littlewood, "Some new convergece criteria for Fourier series" J. London. Math. Soc. , 7 (1932) pp. 252–256
[2] N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)
How to Cite This Entry:
Hardy-Littlewood criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hardy-Littlewood_criterion&oldid=22547
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article