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A restriction on the behaviour of increase of a function. If for any points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059690/l0596901.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059690/l0596902.png" /> belonging to an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059690/l0596903.png" /> the increase of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059690/l0596904.png" /> satisfies the inequality
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A restriction on the behaviour of increase of a function. If for any points $x$ and $x'$ belonging to an interval $[a,b]$ the increase of a function $f$ satisfies the inequality
 +
\begin{equation}\label{eq:1}
 +
|f(x)-f(x')| \leq M|x-x'|^{\alpha},
 +
\end{equation}
  
<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/l059690/l0596905.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
where $0<\alpha\leq1$ and $M$ is a constant, then one says that $f$ satisfies a Lipschitz condition of order $\alpha$ on $[a,b]$ and writes $f\in\operatorname{Lip}\alpha$, $f\in\operatorname{Lip}_M\alpha$ or $f\in H^{\alpha}(M)$. Every function that satisfies a Lipschitz condition with some $\alpha>0$ on $[a,b]$ is uniformly continuous on $[a,b]$, and functions that satisfy a Lipschitz condition of order $\alpha=1$ are absolutely continuous (cf. [[Absolute continuity|Absolute continuity]]; [[Uniform continuity|Uniform continuity]]). A function that has a bounded derivative on $[a,b]$ satisfies a Lipschitz condition on $[a,b]$ with any $\alpha\leq 1$.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059690/l0596906.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059690/l0596907.png" /> is a constant, then one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059690/l0596908.png" /> satisfies a Lipschitz condition of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059690/l05969010.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059690/l05969011.png" /> and writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059690/l05969012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059690/l05969013.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059690/l05969014.png" />. Every function that satisfies a Lipschitz condition with some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059690/l05969015.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059690/l05969016.png" /> is uniformly continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059690/l05969017.png" />, and functions that satisfy a Lipschitz condition of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059690/l05969018.png" /> are absolutely continuous (cf. [[Absolute continuity|Absolute continuity]]; [[Uniform continuity|Uniform continuity]]). A function that has a bounded derivative on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059690/l05969019.png" /> satisfies a Lipschitz condition on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059690/l05969020.png" /> with any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059690/l05969021.png" />.
+
The Lipschitz condition \eqref{eq:1} is equivalent to the condition
 +
\begin{equation}
 +
\omega(\delta,f)\leq M\delta^{\alpha},
 +
\end{equation}
  
The Lipschitz condition (*) is equivalent to the condition
+
where $\omega(\delta,f)$ is the modulus of continuity (cf. [[Continuity, modulus of|Continuity, modulus of]]) of $f$ on $[a,b]$. Lipschitz conditions were first considered by R. Lipschitz [[#References|[1]]] as a sufficient condition for the convergence of the [[Fourier series|Fourier series]] of $f$. In the case $0<\alpha<1$ the condition \eqref{eq:1} is also called a [[Hölder condition|Hölder condition]] of order $\alpha$.
 
 
<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/l059690/l05969022.png" /></td> </tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059690/l05969023.png" /> is the modulus of continuity (cf. [[Continuity, modulus of|Continuity, modulus of]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059690/l05969024.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059690/l05969025.png" />. Lipschitz conditions were first considered by R. Lipschitz [[#References|[1]]] as a sufficient condition for the convergence of the [[Fourier series|Fourier series]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059690/l05969026.png" />. In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059690/l05969027.png" /> the condition (*) is also called a [[Hölder condition|Hölder condition]] of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059690/l05969028.png" />.
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Lipschitz,  "De explicatione per series trigonometricas insttuenda functionum unius variablis arbitrariarum, et praecipue earum, quae per variablis spatium finitum valorum maximorum et minimorum numerum habent infintum disquisitio"  ''J. Reine Angew. Math.'' , '''63'''  (1864)  pp. 296–308  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)  {{MR|0933759}} {{ZBL|0628.42001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.P. Natanson,  "Constructive function theory" , '''1–3''' , F. Ungar  (1964–1965)  (Translated from Russian)  {{MR|1868029}} {{MR|0196342}} {{MR|0196341}} {{MR|0196340}} {{ZBL|1034.01022}} {{ZBL|0178.39701}} {{ZBL|0136.36302}} {{ZBL|0133.31101}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Lipschitz,  "De explicatione per series trigonometricas insttuenda functionum unius variablis arbitrariarum, et praecipue earum, quae per variablis spatium finitum valorum maximorum et minimorum numerum habent infintum disquisitio"  ''J. Reine Angew. Math.'' , '''63'''  (1864)  pp. 296–308  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)  {{MR|0933759}} {{ZBL|0628.42001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.P. Natanson,  "Constructive function theory" , '''1–3''' , F. Ungar  (1964–1965)  (Translated from Russian)  {{MR|1868029}} {{MR|0196342}} {{MR|0196341}} {{MR|0196340}} {{ZBL|1034.01022}} {{ZBL|0178.39701}} {{ZBL|0136.36302}} {{ZBL|0133.31101}} </TD></TR></table>

Revision as of 08:33, 23 November 2012

A restriction on the behaviour of increase of a function. If for any points $x$ and $x'$ belonging to an interval $[a,b]$ the increase of a function $f$ satisfies the inequality \begin{equation}\label{eq:1} |f(x)-f(x')| \leq M|x-x'|^{\alpha}, \end{equation}

where $0<\alpha\leq1$ and $M$ is a constant, then one says that $f$ satisfies a Lipschitz condition of order $\alpha$ on $[a,b]$ and writes $f\in\operatorname{Lip}\alpha$, $f\in\operatorname{Lip}_M\alpha$ or $f\in H^{\alpha}(M)$. Every function that satisfies a Lipschitz condition with some $\alpha>0$ on $[a,b]$ is uniformly continuous on $[a,b]$, and functions that satisfy a Lipschitz condition of order $\alpha=1$ are absolutely continuous (cf. Absolute continuity; Uniform continuity). A function that has a bounded derivative on $[a,b]$ satisfies a Lipschitz condition on $[a,b]$ with any $\alpha\leq 1$.

The Lipschitz condition \eqref{eq:1} is equivalent to the condition \begin{equation} \omega(\delta,f)\leq M\delta^{\alpha}, \end{equation}

where $\omega(\delta,f)$ is the modulus of continuity (cf. Continuity, modulus of) of $f$ on $[a,b]$. Lipschitz conditions were first considered by R. Lipschitz [1] as a sufficient condition for the convergence of the Fourier series of $f$. In the case $0<\alpha<1$ the condition \eqref{eq:1} is also called a Hölder condition of order $\alpha$.

References

[1] R. Lipschitz, "De explicatione per series trigonometricas insttuenda functionum unius variablis arbitrariarum, et praecipue earum, quae per variablis spatium finitum valorum maximorum et minimorum numerum habent infintum disquisitio" J. Reine Angew. Math. , 63 (1864) pp. 296–308
[2] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) MR0933759 Zbl 0628.42001
[3] I.P. Natanson, "Constructive function theory" , 1–3 , F. Ungar (1964–1965) (Translated from Russian) MR1868029 MR0196342 MR0196341 MR0196340 Zbl 1034.01022 Zbl 0178.39701 Zbl 0136.36302 Zbl 0133.31101
How to Cite This Entry:
Lipschitz condition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lipschitz_condition&oldid=28810
This article was adapted from an original article by A.V. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article