View source for Dini criterion

If a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d0325201.png" />-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d0325202.png" /> which is integrable on the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d0325203.png" /> satisfies the condition

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at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d0325205.png" /> for a fixed number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d0325206.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d0325207.png" />, and an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d0325208.png" />, then the Fourier series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d0325209.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d03252010.png" /> converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d03252011.png" />. The criterion was proved by U. Dini [[#References|[1]]]. It is a final (sharp) statement in the following sense. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d03252012.png" /> is a continuous function such that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d03252013.png" /> is not integrable in a neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d03252014.png" />, it is possible to find a continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d03252015.png" /> whose Fourier series diverges at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d03252016.png" /> and such that

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for small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032520/d03252018.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  U. Dini,   "Serie di Fourier e altre rappresentazioni analitiche delle funzioni di una variabile reale" , Pisa  (1880)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Zygmund,   "Trigonometric series" , '''1''' , Cambridge Univ. Press  (1988)</TD></TR></table>