View source for Lipschitz integral condition

A restriction on the behaviour of increase of a function in an integral metric. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059710/l0597101.png" /> in a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059710/l0597102.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059710/l0597103.png" /> satisfies the Lipschitz integral condition of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059710/l0597104.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059710/l0597105.png" /> with constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059710/l0597106.png" /> if

<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/l/l059/l059710/l0597107.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059710/l0597108.png" />. In this case one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059710/l0597109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059710/l05971010.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059710/l05971011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059710/l05971012.png" />. For the case of a periodic function (with period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059710/l05971013.png" />) the Lipschitz integral condition is defined similarly, only in inequality (*) the upper limit of integration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059710/l05971014.png" /> must be replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059710/l05971015.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Zygmund,   "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.M. Nikol'skii,   "Approximation of functions of several variables and imbedding theorems" , Springer  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  O.V. Besov,   V.P. Il'in,   S.M. Nikol'skii,   "Integral representations of functions and imbedding theorems" , Wiley  (1978)  (Translated from Russian)</TD></TR></table>