If a -periodic function which is integrable on the segment satisfies the condition
at a point for a fixed number , , and an arbitrary , then the Fourier series of at converges to . The criterion was proved by U. Dini . It is a final (sharp) statement in the following sense. If is a continuous function such that the function is not integrable in a neighbourhood of the point , it is possible to find a continuous function whose Fourier series diverges at and such that
for small .
|||U. Dini, "Serie di Fourier e altre rappresentazioni analitiche delle funzioni di una variabile reale" , Pisa (1880)|
|||A. Zygmund, "Trigonometric series" , 1 , Cambridge Univ. Press (1988)|
Dini criterion. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Dini_criterion&oldid=16762