View source for Dini-Lipschitz criterion

If a continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032540/d0325401.png" />-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032540/d0325402.png" /> satisfies the condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032540/d0325403.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032540/d0325404.png" /> is the modulus of continuity of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032540/d0325405.png" />, then its Fourier series converges uniformly to it on the entire real axis. The criterion was demonstrated by U. Dini [[#References|[1]]], and also by R. Lipschitz for the special case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032540/d0325406.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032540/d0325407.png" />, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032540/d0325408.png" /> [[#References|[2]]]. The Dini–Lipschitz criterion is a final (sharp) statement in the following sense. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032540/d0325409.png" /> is an arbitrary modulus of continuity satisfying the condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032540/d03254010.png" /></td> </tr></table>

then there exists a continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032540/d03254011.png" />-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032540/d03254012.png" /> whose Fourier series diverges at some point, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032540/d03254013.png" /> satisfies the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032540/d03254014.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  U. Dini,   "Sopra la serie di Fourier" , Pisa  (1872)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Lipschitz,   "De explicatione per series trigonometricas instituenda functionum unius variabilis arbitrariarum, etc."  ''J. Reine Angew. Math.'' , '''63''' :  2  (1864)  pp. 296–308</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Lebesgue,   "Sur la répresentation trigonométrique approchée des fonctions satisfiasants à une condition de Lipschitz"  ''Bull. Soc. Math. France'' , '''38'''  (1910)  pp. 184–210</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.M. Nikol'skii,   "On the Dini–Lipschitz condition for convergence of Fourier series"  ''Doklady Akad. Nauk SSSR'' , '''73''' :  3  (1950)  pp. 457–460  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,   "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)</TD></TR></table>