Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Hölder condition
Summary
In mathematics, a real or complex-valued function f on d-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are real constants C ≥ 0, α > 0, such that
for all x and y in the domain of f. More generally, the condition can be formulated for functions between any two metric spaces. The number α is called the exponent of the Hölder condition. A function on an interval satisfying the condition with α > 1 is constant. If α = 1, then the function satisfies a Lipschitz condition. For any α > 0, the condition implies the function is uniformly continuous. The condition is named after Otto Hölder.
We have the following chain of strict inclusions for functions defined on a closed and bounded interval [a, b] of the real line with a < b :
Continuously differentiable ⊂ Lipschitz continuous ⊂ α-Hölder continuous ⊂ uniformly continuous ⊂ continuous,
where 0 < α ≤ 1.
Hölder spaces consisting of functions satisfying a Hölder condition are basic in areas of functional analysis relevant to solving partial differential equations, and in dynamical systems. The Hölder space Ck,α(Ω), where Ω is an open subset of some Euclidean space and k ≥ 0 an integer, consists of those functions on Ω having continuous derivatives up through order k and such that the kth partial derivatives are Hölder continuous with exponent α, where 0 < α ≤ 1. This is a locally convex topological vector space. If the Hölder coefficient
is finite, then the function f is said to be (uniformly) Hölder continuous with exponent α in Ω. In this case, the Hölder coefficient serves as a seminorm. If the Hölder coefficient is merely bounded on compact subsets of Ω, then the function f is said to be locally Hölder continuous with exponent α in Ω.
If the function f and its derivatives up to order k are bounded on the closure of Ω, then the Hölder space can be assigned the norm
where β ranges over multi-indices and
These seminorms and norms are often denoted simply and or also and in order to stress the dependence on the domain of f.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.