Difference between revisions of "Hardy-Littlewood criterion"
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''for the convergence of a Fourier series'' | ''for the convergence of a Fourier series'' | ||
| − | If a | + | 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 | + | 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 | + | 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]]]. | The criterion was established by G.H. Hardy and J.E. Littlewood [[#References|[1]]]. | ||
Latest revision as of 17:25, 19 September 2014
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) |
How to Cite This Entry:
Hardy-Littlewood criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hardy-Littlewood_criterion&oldid=22547
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