Hardy-Littlewood criterion

for the convergence of a Fourier series

If a $2\pi$-periodic function $f$ is such that

$$f(x_0+h)-f(x_0)=o\left(\frac{1}{\log1/|h|}\right),\quad|h|\to+0,$$

and if its Fourier coefficients $a_n,b_n$ satisfy the conditions

$$a_n,b_n=O(n^{-\delta}),\quad n\to+\infty,$$

for some $\delta>0$, then the Fourier series of $f$ at $x_0$ converges to $f(x_0)$.

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)