Hölder condition

Hölder condition

In mathematics, a real-valued function "f" on R"n" satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants "C", α, such that, forall x, y in mathbf{R}^n ,

: | f(x) - f(y) | leq C |x - y| ^{alpha}.

This condition generalizes to functions between any two metric spaces. The number α is called the "exponent" of the Hölder condition. If alpha=1, then the function satisfies a Lipschitz condition. If alpha=0, then the function simply is bounded.

Hölder spaces consisting of functions satisfying a Hölder condition are basic in areas of functional analysis relevant to solving partial differential equations. The Hölder space C^{n, alpha} (Omega), where Ω is an open subset of some Euclidean space, consists of those functions whose derivatives up to order "n" are Hölder continuous with exponent α. This is a topological vector space, with a seminorm

: | f |_{C^{0,alpha = sup_{x,y in Omega} frac{|x-y|^alpha},

and for ngeq 0 the norm is given by

: | f |_{C^{n, alpha = |f|_{C^n}+max_{| eta | = n} | D^eta f |_{C^{0,alphawhere β ranges over multi-indices and:|f|_{C^n}=max_{| eta | leq n}sup_{xinOmega} |D^eta f (x)|

Examples in C^{0,alpha}({mathbb R})

* If 0 then all C^{0,eta} Hölder continuous functions are also C^{0,alpha} Hölder continuous. This also includes eta=1 and therefore all Lipschitz continuous functions are also C^{0,alpha} Hölder continuous.

* The function f(x)=sqrt{x} defined on [0,3] is "not" Lipschitz continuous, but is C^{0,alpha} Hölder continuous for alphalefrac12.

References


Wikimedia Foundation. 2010.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

  • Hölder condition — /hel deuhr/; Ger. /huel deuhrdd/, Math. See Lipschitz condition. [after German mathematician Ludwig Otto Hölder (1859 1937)] * * * …   Universalium

  • Hölder condition — /hel deuhr/; Ger. /huel deuhrdd/, Math. See Lipschitz condition. [after German mathematician Ludwig Otto Hölder (1859 1937)] …   Useful english dictionary

  • Holder — may refer to:People with the surname Holder: * Holder (surname)In mathematics: * Hölder condition * Hölder s inequality * Hölder mean * Jordan Hölder theoremIn sports: * Holder (American football)In devices: * Cup holder * Flame holder, a jet… …   Wikipedia

  • Condition subsequent — refers to an event or state of affairs that brings an end to something else. A condition subsequent is often used in a legal context as a marker bringing an end to one s legal rights or duties. A condition subsequent may be either an event or a… …   Wikipedia

  • Condition Critical — Album par Quiet Riot Sortie 27 Juillet 1984 Enregistrement 1984 Durée 44:02 Genre Heavy Metal Producteur Spen …   Wikipédia en Français

  • Condition Critical — Álbum de Quiet Riot Publicación 27 de Julio de 1984 Grabación 1984 Género(s) Heavy metal, Glam metal …   Wikipedia Español

  • Condition de Hölder — En analyse, la continuité höldérienne ou la condition de Hölder est une condition suffisante (mais non nécessaire) pour qu’une application définie entre deux espaces métriques soit continue. La définition s’applique en particulier pour les… …   Wikipédia en Français

  • Condition Critical — Winners Take All redirects here. For the album, see Winners Take All (album). Condition Critical Studio album by Quiet Riot …   Wikipedia

  • Otto Hölder — Otto Ludwig Hölder Otto Hölder Born 22 December 1859(1859 12 22) Stuttgart, Germany …   Wikipedia

  • Lipschitz condition — /lip shits/, Math. the property of a function on a closed interval such that the absolute value of the difference in functional values at any two points in the interval is less than a constant times the absolute value of the difference of the… …   Universalium

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”