Hardy-Littlewood criterion

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)