View source for Hardy-Littlewood criterion

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''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 [[#References|[1]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.H. Hardy,   J.E. Littlewood,   "Some new convergece criteria for Fourier series"  ''J. London. Math. Soc.'' , '''7'''  (1932)  pp. 252–256</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,   "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)</TD></TR></table>