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Difference between revisions of "Dini criterion"

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then the Fourier series of $f$ converges to $S$ at $x$.  
 
then the Fourier series of $f$ converges to $S$ at $x$.  
  
Cp. with Section 38 of Chapter I in volume 1 of {{Cite|Ba}} and Section 6 of Chapter II in volume 1 of {{Cite|Zy}}. Observe that, if \eqref{e:Dini} holds, then the right and left limits $f (x^+)$ and $f(x^-)$ of $f$ at $x$ exists and $S= \frac{f(x^+)+f(x^-)}{2}$.
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Cp. with Section 38 of Chapter I in volume 1 of {{Cite|Ba}} and Section 6 of Chapter II in volume 1 of {{Cite|Zy}}. Observe that, if \eqref{e:Dini} holds and in addition the limit $\lim_{t\downarrow0}[f(x+t)+f(x-t)]$ exists, then $S=\lim_{t\downarrow0}[f(x+t)+f(x-t)]$. {{Cite|Di}}  
  
 
From Dini's statement it is possible to conclude several classical corollaries, for instance  
 
From Dini's statement it is possible to conclude several classical corollaries, for instance  

Latest revision as of 15:51, 27 June 2017

for the convergence of Fourier series

2010 Mathematics Subject Classification: Primary: 42A20 [MSN][ZBL]

A criterion first proved by Dini for the convergence of Fourier series in [Di].

Theorem Consider a summable $2\pi$ periodic function $f$ and a point $x\in \mathbb R$. If there is a number $S$ and a $\delta>0$ such that \begin{equation}\label{e:Dini} \int_0^\delta |f(x+u) + f(x-u)-2S| \frac{du}{u} < \infty \end{equation} then the Fourier series of $f$ converges to $S$ at $x$.

Cp. with Section 38 of Chapter I in volume 1 of [Ba] and Section 6 of Chapter II in volume 1 of [Zy]. Observe that, if \eqref{e:Dini} holds and in addition the limit $\lim_{t\downarrow0}[f(x+t)+f(x-t)]$ exists, then $S=\lim_{t\downarrow0}[f(x+t)+f(x-t)]$. [Di]

From Dini's statement it is possible to conclude several classical corollaries, for instance

  • the convergence of the Fourier series of $f$ to $f(x)$ at every point where $f$ is differentiable
  • the convergence of the Fourier series of $f$ to $f$ when $f$ is Hölder continuous.

It is also a (sharp) statement in the following sense. If $\omega: ]0, \infty[\to ]0, \infty[$ is a continuous function such that $\frac{\omega (t)}{t}$ is not integrable in a neighborhood of the origin, then there is a continuous $2\pi$-periodic function $f:\mathbb R \to \mathbb R$ such that $|f(t)-f(0)|\leq \omega (|t|)$ for every $t$ and the Fourier series of $f$ diverges at $0$.

The Dini criterion is weaker then the De la Vallee-Poussin criterion and not comparable to the Jordan criterion, cp. with Sections 2 and 3 of Chapter III in [Ba].

References

[Ba] N.K. Bary, "A treatise on trigonometric series" , Pergamon, 1964.
[Di] U. Dini, "Serie di Fourier e altre rappresentazioni analitiche delle funzioni di una variabile reale" , Pisa (1880).
[Ed] R. E. Edwards, "Fourier series". Vol. 1. Holt, Rineheart and Winston, 1967.
[Zy] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) MR0933759 Zbl 0628.42001
How to Cite This Entry:
Dini criterion. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Dini_criterion&oldid=41650
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article