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If a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d0325201.png" />-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d0325202.png" /> which is integrable on the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d0325203.png" /> satisfies the condition
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''for the convergence of Fourier series''
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d0325204.png" /></td> </tr></table>
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{{MSC|42A20}}
  
at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d0325205.png" /> for a fixed number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d0325206.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d0325207.png" />, and an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d0325208.png" />, then the Fourier series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d0325209.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d03252010.png" /> converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d03252011.png" />. The criterion was proved by U. Dini [[#References|[1]]]. It is a final (sharp) statement in the following sense. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d03252012.png" /> is a continuous function such that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d03252013.png" /> is not integrable in a neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d03252014.png" />, it is possible to find a continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d03252015.png" /> whose Fourier series diverges at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d03252016.png" /> and such that
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d03252017.png" /></td> </tr></table>
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A criterion first proved by Dini for the convergence of Fourier series in {{Cite|Di}}.  
  
for small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d03252018.png" />.
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'''Theorem'''
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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 {{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 $S= \frac{f(x^+)+f(x^-)}{2}$.
 +
 
 +
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 H\"older 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 {{Cite|Ba}}.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> U. Dini,  "Serie di Fourier e altre rappresentazioni analitiche delle funzioni di una variabile reale" , Pisa  (1880)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"A. Zygmund,  "Trigonometric series" , '''1''' , Cambridge Univ. Press  (1988)</TD></TR></table>
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{|
 +
|-
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|valign="top"|{{Ref|Ba}}|| N.K. Bary,  "A treatise on trigonometric series" , Pergamon, 1964.
 +
|-
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|valign="top"|{{Ref|Di}}|| U. Dini,  "Serie di Fourier e altre rappresentazioni analitiche delle funzioni di una variabile reale" , Pisa  (1880).
 +
|-
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|valign="top"|{{Ref|Ed}}|| R. E. Edwards, "Fourier series". Vol. 1. Holt, Rineheart and Winston, 1967.
 +
|-
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|valign="top"|{{Ref|Zy}}||  A. Zygmund,  "Trigonometric series" , '''1–2''' , Cambridge Univ.   Press  (1988) {{MR|0933759}}  {{ZBL|0628.42001}}
 +
|-
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|}

Revision as of 20:23, 16 October 2012

for the convergence of Fourier series

2020 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, then the right and left limits $f (x^+)$ and $f(x^-)$ of $f$ at $x$ exists and $S= \frac{f(x^+)+f(x^-)}{2}$.

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 H\"older 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://encyclopediaofmath.org/index.php?title=Dini_criterion&oldid=16762
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article