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One of the sufficiency criteria for the convergence of a [[Fourier series|Fourier series]] at a point. Let a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099060/y0990601.png" /> have period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099060/y0990602.png" />, be Lebesgue integrable over the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099060/y0990603.png" /> and, on putting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099060/y0990604.png" />, let it satisfy at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099060/y0990605.png" /> the conditions: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099060/y0990606.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099060/y0990607.png" />; 2) the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099060/y0990608.png" /> is of finite variation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099060/y0990609.png" /> on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099060/y09906010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099060/y09906011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099060/y09906012.png" /> is some fixed number; and 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099060/y09906013.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099060/y09906014.png" />. Then the Fourier series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099060/y09906015.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099060/y09906016.png" /> converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099060/y09906017.png" /> (cf. [[#References|[2]]]). Young's criterion is stronger than the [[Jordan criterion|Jordan criterion]]. It was established by W.H. Young [[#References|[1]]].
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''for the convergence of Fourier series''
  
====References====
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{{MSC|42A20}}
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W.H. Young,  "On the convergence of the derived series of Fourier series"  ''Proc. London Math. Soc.'' , '''17'''  (1916)  pp. 195–236</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>
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{{TEX|done}}
  
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A criterion first proved by W. H. Young for the convergence of Fourier series in {{Cite|Yo}}.
  
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'''Theorem'''
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Consider a summable $2\pi$ periodic function $f$, a point $x\in \mathbb R$ and the function
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\[
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\varphi (u):= f(x+u)+f(x-u) - 2 f(x)
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\]
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Assume that:
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* $\varphi (u)\to 0$ as $u\downarrow 0$;
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* $\theta (t) = t\varphi (t)$ is a function of bounded variation in some interval $]0, \delta[$
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* the total variation of $\theta$ on the interval $]0, h[$ is $O(h)$.
 +
Then the Fourier series of $f$ converges to $f(x)$ at $x$.
  
====Comments====
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Cp. with Section 4 of Chapter III in volume 1 of {{Cite|Ba}}.
  
 +
The Young's criterion is stronger than the [[Dirichlet theorem|Dirichlet criterion]], the [[Dini criterion]] and the [[Jordan criterion]], it is not comparable to the [[De la Vallee-Poussin criterion]] and it is weaker than the [[Lebesgue criterion]]. Cp. with Sections 5 and 7 of Chapter III in volume 1 of {{Cite|Ba}}.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Zygmund,  "Trigonometric series" , '''2''' , Cambridge Univ. Press  (1988)</TD></TR></table>
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{|
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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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|-
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|valign="top"|{{Ref|Ed}}|| R. E. Edwards, "Fourier series". Vol. 1. Holt, Rineheart and Winston, 1967.
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|-
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|valign="top"|{{Ref|Yo}}|| W.H. Young,  "On the convergence of the derived series of Fourier  series"  ''Proc. London Math. Soc.'' , '''17'''  (1916)  pp. 195–236
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|-
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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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|-
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|}

Latest revision as of 20:45, 16 October 2012

for the convergence of Fourier series

2020 Mathematics Subject Classification: Primary: 42A20 [MSN][ZBL]

A criterion first proved by W. H. Young for the convergence of Fourier series in [Yo].

Theorem Consider a summable $2\pi$ periodic function $f$, a point $x\in \mathbb R$ and the function \[ \varphi (u):= f(x+u)+f(x-u) - 2 f(x) \] Assume that:

  • $\varphi (u)\to 0$ as $u\downarrow 0$;
  • $\theta (t) = t\varphi (t)$ is a function of bounded variation in some interval $]0, \delta[$
  • the total variation of $\theta$ on the interval $]0, h[$ is $O(h)$.

Then the Fourier series of $f$ converges to $f(x)$ at $x$.

Cp. with Section 4 of Chapter III in volume 1 of [Ba].

The Young's criterion is stronger than the Dirichlet criterion, the Dini criterion and the Jordan criterion, it is not comparable to the De la Vallee-Poussin criterion and it is weaker than the Lebesgue criterion. Cp. with Sections 5 and 7 of Chapter III in volume 1 of [Ba].

References

[Ba] N.K. Bary, "A treatise on trigonometric series" , Pergamon, 1964.
[Ed] R. E. Edwards, "Fourier series". Vol. 1. Holt, Rineheart and Winston, 1967.
[Yo] W.H. Young, "On the convergence of the derived series of Fourier series" Proc. London Math. Soc. , 17 (1916) pp. 195–236
[Zy] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) MR0933759 Zbl 0628.42001
How to Cite This Entry:
Young criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Young_criterion&oldid=19222
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article