In mathematics, Dini's criterion is a condition for the pointwise convergence of Fourier series, introduced by Ulisse Dini (1880).
Statement
Dini's criterion states that if a periodic function has the property that is locally integrable near , then the Fourier series of converges to at .
Dini's criterion is in some sense as strong as possible: if is a positive continuous function such that is not locally integrable near , there is a continuous function with whose Fourier series does not converge at .
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) , "Dini criterion", Encyclopedia of Mathematics, EMS Press