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Jordan criterion

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for the convergence of Fourier series

2010 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 JFM Zbl 13.0184.01
[Zy] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) MR0933759 Zbl 0628.42001
How to Cite This Entry:
Jordan criterion. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Jordan_criterion&oldid=28288
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article