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Difference between revisions of "Jordan criterion"

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''for the convergence of Fourier series''
 
''for the convergence of Fourier series''
  
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If a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054290/j0542901.png" />-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054290/j0542902.png" /> has bounded variation on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054290/j0542903.png" />, then its Fourier series converges at every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054290/j0542904.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054290/j0542905.png" />; if, in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054290/j0542906.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054290/j0542907.png" />, then its Fourier series converges to it uniformly on every interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054290/j0542908.png" /> strictly inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054290/j0542909.png" />. The criterion was established by C. Jordan [[#References|[1]]]; it generalizes the [[Dirichlet theorem|Dirichlet theorem]] on the convergence of Fourier series of piecewise-monotone functions.
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{{MSC|42A20}}
  
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====References====
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{{TEX|done}}
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Jordan,  "Sur la série de Fourier"  ''C.R. Acad. Sci. Paris'' , '''92'''  (1881)  pp. 228–230</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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[[Category:Harmonic analysis on euclidean spaces]]
  
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A criterion first proved by Jordan for the convergence of Fourier series in {{Cite|Jo}}. The criterion, which generalizes the [[Dirichlet theorem]] on the convergence of Fourier series of piecewise monotone functions, is also called Dirichlet-Jordan test, cf. with {{Cite|Zy}}.
  
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'''Theorem'''
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Let $f: \mathbb R\to\mathbb R$ be a $2\pi$ periodic summable function.
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*  If $f$ has bounded variation in an open interval $I$ then its Fourier  series converges to $\frac{1}{2} (f (x^+) + f(x^-))$ at every $x\in I$.
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*  If in addition $f$ is continuous in $I$ then its Fourier series  converges uniformly to $f$ on every closed interval $J\subset I$.
  
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====Comments====
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For a proof see Section 10.1 and Exercises 10.13 and 10.14 of {{Cite|Ed}}.
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The Jordan criterion is also called the Dirichlet–Jordan test, cf. [[#References|[a1]]].
 
  
 
====References====
 
====References====
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Zygmund,  "Trigonometric series" , '''1–2''' , 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|Jo}}|| C. Jordan,  "Sur la série de Fourier" ''C.R. Acad. Sci. Paris'' , '''92'''  (1881)  pp. 228–230
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|-
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|valign="top"|{{Ref|Zy}}|| A. Zygmund,  "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)
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|-
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|}

Revision as of 18:56, 20 August 2012

for the convergence of Fourier series

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

A criterion first proved by Jordan for the convergence of Fourier series in [Jo]. The criterion, which generalizes the Dirichlet theorem on the convergence of Fourier series of piecewise monotone functions, is also called Dirichlet-Jordan test, cf. with [Zy].

Theorem Let $f: \mathbb R\to\mathbb R$ be a $2\pi$ periodic summable function.

  • If $f$ has bounded variation in an open interval $I$ then its Fourier series converges to $\frac{1}{2} (f (x^+) + f(x^-))$ at every $x\in I$.
  • If in addition $f$ is continuous in $I$ then its Fourier series converges uniformly to $f$ on every closed interval $J\subset I$.

For a proof see Section 10.1 and Exercises 10.13 and 10.14 of [Ed].

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.
[Jo] C. Jordan, "Sur la série de Fourier" C.R. Acad. Sci. Paris , 92 (1881) pp. 228–230
[Zy] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)
How to Cite This Entry:
Jordan criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jordan_criterion&oldid=27695
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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