Namespaces
Variants
Actions

Difference between revisions of "Jordan criterion"

From Encyclopedia of Mathematics
Jump to: navigation, search
m
m
Line 7: Line 7:
 
[[Category:Harmonic analysis on Euclidean spaces]]
 
[[Category:Harmonic analysis on Euclidean spaces]]
  
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}}.
+
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 the [[Trigonometric series|Fourier series]] of piecewise monotone functions, is also called Dirichlet-Jordan test, cf. with {{Cite|Zy}}.
  
 
'''Theorem'''
 
'''Theorem'''

Revision as of 13:12, 22 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 the 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=27696
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article