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Difference between revisions of "Lipschitz constant"

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''for a function $f$ defined on an interval $[a,b]$''
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{{MSC|54E40}}
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[[Category:Analysis]]
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{{TEX|done}}
  
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The greatest lower bound of constants $M>0$ in the [[Lipschitz condition|Lipschitz condition]] of order $\alpha$, $0<\alpha\leq1$,
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''of a function $f$''
  
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====Definition====
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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*}
 
\begin{equation*}
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|f(y)-f(x)|\leq M|y-x|^{\alpha}, \quad x,y\in[a,b].
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\sup_{x\neq y} \frac{|f(y)-f(x)|}{|y-x|}\, .
 
\end{equation*}
 
\end{equation*}
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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]]).
  
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====Relations with differentiability====
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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
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\[
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\sup_x |f'(x)|\, .
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\]
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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.
  
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In other words, one can find
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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 $S$.
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\begin{equation*}
 
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M = \sup\limits_{x,y\in[a,b]}\frac{|f(y)-f(x)|}{|y-x|^{\alpha}}.
 
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\end{equation*}
 
  
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If $\alpha=1$ $f$ is called [[Lipschitz Function | Lipschitz function]].
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====Remark on terminology====
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The one used here is the most common terminology. However, some authors use the name Lipschitz condition of exponent $\alpha$ for the [[Hölder condition]]
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\begin{equation}\label{eq:1}
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|f(y)-f(x)| \leq M |y-x|^{\alpha}\,
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\end{equation}
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(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:
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* ''Hölder exponent'' for the fixed exponent $\alpha$
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*''Hölder constant'' for the smallest $M$ satisfying \eqref{eq:1}.
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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=30687
This article was adapted from an original article by A.V. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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