In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a real or complex-valued function ''f'' on ''d''-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants ''C'', 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 space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s. 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
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exi ...
. For any α > 0, the condition implies the function is
uniformly continuous
In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
. The condition is named after
Otto Hölder
Ludwig Otto Hölder (December 22, 1859 – August 29, 1937) was a German mathematician born in Stuttgart.
Early life and education
Hölder was the youngest of three sons of professor Otto Hölder (1811–1890), and a grandson of professor Chris ...
.
We have the following chain of strict inclusions for functions over a closed and bounded non-trivial interval of the real line:
:
Continuously differentiable ⊂
Lipschitz continuous
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there e ...
⊂ α-Hölder continuous ⊂
uniformly continuous
In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
⊂
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
,
where 0 < α ≤ 1.
Hölder spaces
Hölder spaces consisting of functions satisfying a Hölder condition are basic in areas of
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
relevant to solving
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
, and in
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
s. The Hölder space ''C''
''k'',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
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
s up through order ''k'' and such that the ''k''th partial derivatives are Hölder continuous with exponent α, where 0 < α ≤ 1. This is a
locally convex topological vector space
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
. 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 In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk ...
. If the Hölder coefficient is merely bounded on
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
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
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distribution (mathematics), distributions, by generalising the concept of an integer index not ...
and
:
These seminorms and norms are often denoted simply
and
or also
and
in order to stress the dependence on the domain of ''f''. If Ω is open and bounded, then
is a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
with respect to the norm
.
Compact embedding of Hölder spaces
Let Ω be a bounded subset of some Euclidean space (or more generally, any totally bounded metric space) and let 0 < α < β ≤ 1 two Hölder exponents. Then, there is an obvious inclusion map of the corresponding Hölder spaces:
:
which is continuous since, by definition of the Hölder norms, we have:
:
Moreover, this inclusion is compact, meaning that bounded sets in the ‖ · ‖
0,β norm are relatively compact in the ‖ · ‖
0,α norm. This is a direct consequence of the
Ascoli-Arzelà theorem. Indeed, let (''u
n'') be a bounded sequence in ''C''
0,β(Ω). Thanks to the Ascoli-Arzelà theorem we can assume without loss of generality that ''u
n'' → ''u'' uniformly, and we can also assume ''u'' = 0. Then
:
because
:
Examples
* If 0 < α ≤ β ≤ 1 then all
Hölder continuous functions on a ''bounded set'' Ω are also
Hölder continuous. This also includes β = 1 and therefore all
Lipschitz continuous
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there e ...
functions on a bounded set are also ''C''
0,α Hölder continuous.
* The function ''f''(''x'') = ''x''
β (with β ≤ 1) defined on
, 1serves as a prototypical example of a function that is ''C''
0,α Hölder continuous for 0 < α ≤ β, but not for α > β. Further, if we defined ''f'' analogously on
, it would be ''C''
0,α Hölder continuous only for α = β.
* For α > 1, any α–Hölder continuous function on
, 1(or any interval) is a constant.
* There are examples of uniformly continuous functions that are not α–Hölder continuous for any α. For instance, the function defined on [0, 1/2] by ''f''(0) = 0 and by ''f''(''x'') = 1/log(''x'') otherwise is continuous, and therefore uniformly continuous by the Heine-Cantor theorem. It does not satisfy a Hölder condition of any order, however.
*The
Weierstrass function
In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass.
The Weierstr ...
defined by:
::
:where