Difference between revisions of "Lipschitz constant"
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| − | + | {{MSC|54E40}} | |
| + | [[Category:Analysis]] | ||
| + | {{TEX|done}} | ||
| − | + | ''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*} | \begin{equation*} | ||
| − | |f(y)-f(x)| | + | \sup_{x\neq y} \frac{|f(y)-f(x)|}{|y-x|}\, . |
\end{equation*} | \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]]). | ||
| − | If $\ | + | ====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 $[f]_\alpha$.
Lipschitz constant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lipschitz_constant&oldid=28851