Jordan criterion

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 JFM Zbl 13.0184.01
[Zy]	A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) MR0933759 Zbl 0628.42001