Namespaces
Variants
Actions

Difference between revisions of "Lipschitz constant"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
 
(2 intermediate revisions by one other user not shown)
Line 1: Line 1:
''for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059700/l0597001.png" /> defined on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059700/l0597002.png" />''
+
{{MSC|54E40}}
 +
[[Category:Analysis]]
 +
{{TEX|done}}
  
The greatest lower bound of constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059700/l0597003.png" /> in the [[Lipschitz condition|Lipschitz condition]] of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059700/l0597004.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059700/l0597005.png" />,
+
''of a function $f$''
  
<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/l059700/l0597006.png" /></td> </tr></table>
+
====Definition====
 +
For functions $f:[a,b]\to \mathbb R$ it denotes the smallest constant $M>0$ in the [[Lipschitz condition]], namely the nonnegative number
 +
\begin{equation*}
 +
\sup_{x\neq y} \frac{|f(y)-f(x)|}{|y-x|}\, .
 +
\end{equation*}
 +
The definition can be readily extended to vector-valued functions on subsets of any Euclidean space and, more in general, to mappings between metric spaces (cf. [[Lipschitz function]]).
 +
 
 +
====Relations with differentiability====
 +
If the domain of $f$ is an interval, the function is everywhere differentiable and the derivative is bounded, then it is easy to see that the Lipschitz constant of $f$ equals
 +
\[
 +
\sup_x |f'(x)|\, .
 +
\]
 +
A corresponding statement holds for vector-valued maps on convex euclidean domains: in this case $|f'(x)|$ must be replaced by $\|Df (x)\|_o$, where $Df$ denotes the Jacobian matrix of the differential of $f$ and $\|\cdot\|_o$ the operator norm.
 +
 
 +
A partial converse of this statement is given by [[Rademacher theorem]]: a Lipschitz function $f$ on an open subset of a Euclidean space is almost everywhere differentiable and the Lipschitz constant bounds from above the number $S = \sup_x \|Df (x)\|_o$, where the supremum is taken over the points $x$ of differentiability. If in addition the domain is convex, we again conclude that the Lipschitz constant equals $S$.
 +
 
 +
====Remark on terminology====
 +
The one used here is the most common terminology. However, some authors use the name Lipschitz condition of exponent $\alpha$ for the [[Hölder condition]]
 +
\begin{equation}\label{eq:1}
 +
|f(y)-f(x)| \leq M |y-x|^{\alpha}\,
 +
\end{equation}
 +
(cp. with [[Lipschitz condition]]). Consistently, such authors use the term Lipschitz constant for the smallest $M$ satisfying \eqref{eq:1} even when $\alpha<1$. For functions satisfying \eqref{eq:1} the most common terminology is instead:
 +
* ''Hölder exponent'' for the fixed exponent $\alpha$
 +
*''Hölder constant'' for the smallest $M$ satisfying \eqref{eq:1}.
 +
A common notation for the latter is $[f]_\alpha$.

Latest revision as of 16:16, 9 November 2013

2020 Mathematics Subject Classification: Primary: 54E40 [MSN][ZBL]

of a function $f$

Definition

For functions $f:[a,b]\to \mathbb R$ it denotes the smallest constant $M>0$ in the Lipschitz condition, namely the nonnegative number \begin{equation*} \sup_{x\neq y} \frac{|f(y)-f(x)|}{|y-x|}\, . \end{equation*} The definition can be readily extended to vector-valued functions on subsets of any Euclidean space and, more in general, to mappings between metric spaces (cf. Lipschitz function).

Relations with differentiability

If the domain of $f$ is an interval, the function is everywhere differentiable and the derivative is bounded, then it is easy to see that the Lipschitz constant of $f$ equals \[ \sup_x |f'(x)|\, . \] A corresponding statement holds for vector-valued maps on convex euclidean domains: in this case $|f'(x)|$ must be replaced by $\|Df (x)\|_o$, where $Df$ denotes the Jacobian matrix of the differential of $f$ and $\|\cdot\|_o$ the operator norm.

A partial converse of this statement is given by Rademacher theorem: a Lipschitz function $f$ on an open subset of a Euclidean space is almost everywhere differentiable and the Lipschitz constant bounds from above the number $S = \sup_x \|Df (x)\|_o$, where the supremum is taken over the points $x$ of differentiability. If in addition the domain is convex, we again conclude that the Lipschitz constant equals $S$.

Remark on terminology

The one used here is the most common terminology. However, some authors use the name Lipschitz condition of exponent $\alpha$ for the Hölder condition \begin{equation}\label{eq:1} |f(y)-f(x)| \leq M |y-x|^{\alpha}\, \end{equation} (cp. with Lipschitz condition). Consistently, such authors use the term Lipschitz constant for the smallest $M$ satisfying \eqref{eq:1} even when $\alpha<1$. For functions satisfying \eqref{eq:1} the most common terminology is instead:

  • Hölder exponent for the fixed exponent $\alpha$
  • Hölder constant for the smallest $M$ satisfying \eqref{eq:1}.

A common notation for the latter is $[f]_\alpha$.

How to Cite This Entry:
Lipschitz constant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lipschitz_constant&oldid=13226
This article was adapted from an original article by A.V. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article