# Difference between revisions of "Jordan criterion"

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 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].

How to Cite This Entry:
Jordan criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jordan_criterion&oldid=17285
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article