Dini criterion
In mathematics, Dini's criterion is a condition for the pointwise convergence of Fourier series, introduced by Dini (1880).
Statement
Dini's criterion states that if a periodic function f has the property that (f(t) + f(–t))/t is locally integrable near 0, then the Fourier series of f converges to 0 at t = 0.
Dini's criterion is in some sense as strong as possible: if g(t) is a positive continuous function such that g(t)/t is not locally integrable near 0, there is a continuous function f with |f(t)| ≤ g(t) whose Fourier series does not converge at 0.
References
- Dini, Ulisse (1880), Serie di Fourier e altre rappresentazioni analitiche delle funzioni di una variabile reale, Pisa: Nistri, ISBN 978-1429704083
- Golubov, B. I. (2001) [1994], "Dini criterion", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
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