HOME

TheInfoList



OR:

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 : , f(x) - f(y) , \leq C\, x - y\, ^ 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 : , f , _ = \sup_ \frac, 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 C^(\overline) can be assigned the norm : \, f \, _ = \, f\, _+\max_ \left , D^\beta f \right , _ 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 :\, f\, _ = \max_ \sup_ \left , D^\beta f (x) \right , . These seminorms and norms are often denoted simply , f , _ and \, f \, _ or also , f , _\; and \, f \, _ in order to stress the dependence on the domain of ''f''. If Ω is open and bounded, then C^(\overline) 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 \, \cdot\, _ .


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: :C^(\Omega)\to C^(\Omega), which is continuous since, by definition of the Hölder norms, we have: :\forall f \in C^(\Omega): \qquad , f , _\le \mathrm(\Omega)^ , f , _. 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 (''un'') be a bounded sequence in ''C''0,β(Ω). Thanks to the Ascoli-Arzelà theorem we can assume without loss of generality that ''un'' → ''u'' uniformly, and we can also assume ''u'' = 0. Then :, u_n-u, _=, u_n, _\to 0, because :\frac = \left(\frac\right)^ \left , u_n(x)-u_n(y) \right , ^ \le , u_n, _^ \left(2\, u_n\, _\infty\right)^=o(1).


Examples

* If 0 < α ≤ β ≤ 1 then all C^(\overline) Hölder continuous functions on a ''bounded set'' Ω are also C^(\overline) 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 [0,\infty), 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: ::f(x)=\sum_^ a^n\cos \left (b^n \pi x \right ), :where 0 is an integer, b \geq 2 and ab>1+\tfrac, is α-Hölder continuous with ::\alpha=-\frac. * The
Cantor function In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in analysis, because it challenges naive intuitions about continuity, derivative, and measure. ...
is Hölder continuous for any exponent \alpha \le \tfrac, and for no larger one. In the former case, the inequality of the definition holds with the constant ''C'' := 2. *
Peano curve In geometry, the Peano curve is the first example of a space-filling curve to be discovered, by Giuseppe Peano in 1890. Peano's curve is a surjective, continuous function from the unit interval onto the unit square, however it is not injecti ...
s from , 1onto the square , 1sup>2 can be constructed to be 1/2–Hölder continuous. It can be proved that when \alpha > \tfrac the image of a α–Hölder continuous function from the unit interval to the square cannot fill the square. * Sample paths of
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
are almost surely everywhere locally α-Hölder for every \alpha < \tfrac. *Functions which are locally integrable and whose integrals satisfy an appropriate growth condition are also Hölder continuous. For example, if we let ::u_ = \frac \int_ u(y) dy :and ''u'' satisfies ::\int_ \left , u(y) - u_ \right , ^2 dy \leq C r^, :then ''u'' is Hölder continuous with exponent α.See, for example, Han and Lin, Chapter 3, Section 1. This result was originally due to
Sergio Campanato Sergio Campanato (17 February 1930 – 1 March 2005) was an Italian mathematician who studied the theory of regularity for elliptic and parabolic partial differential equations. Career He graduated in mathematics and physics at the Universit ...
.
*Functions whose ''oscillation'' decay at a fixed rate with respect to distance are Hölder continuous with an exponent that is determined by the rate of decay. For instance, if ::w(u,x_0,r) = \sup_ u - \inf_ u :for some function ''u''(''x'') satisfies ::w \left (u,x_0,\tfrac \right ) \leq \lambda w \left (u,x_0,r \right ) :for a fixed λ with 0 < λ < 1 and all sufficiently small values of ''r'', then ''u'' is Hölder continuous. *Functions in
Sobolev space In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
can be embedded into the appropriate Hölder space via Morrey's inequality if the spatial dimension is less than the exponent of the Sobolev space. To be precise, if n < p \leq \infty then there exists a constant ''C'', depending only on ''p'' and ''n'', such that: ::\forall u \in C^1 (\mathbf^n) \cap L^p (\mathbf^n): \qquad \, u\, _\leq C \, u\, _, :where \gamma = 1 - \tfrac. Thus if ''u'' ∈ ''W''1, ''p''(R''n''), then ''u'' is in fact Hölder continuous of exponent γ, after possibly being redefined on a set of measure 0.


Properties

*A closed additive subgroup of an infinite dimensional Hilbert space ''H'', connected by α–Hölder continuous arcs with α > 1/2, is a linear subspace. There are closed additive subgroups of ''H'', not linear subspaces, connected by 1/2–Hölder continuous arcs. An example is the additive subgroup ''L''2(R, Z) of the Hilbert space ''L''2(R, R). *Any α–Hölder continuous function ''f'' on a metric space ''X'' admits a Lipschitz approximation by means of a sequence of functions (''fk'') such that ''fk'' is ''k''-Lipschitz and ::\, f-f_k\, _=O \left (k^ \right ). :Conversely, any such sequence (''fk'') of Lipschitz functions converges to an α–Hölder continuous uniform limit ''f''. *Any α–Hölder function ''f'' on a subset ''X'' of a normed space ''E'' admits a uniformly continuous extension to the whole space, which is Hölder continuous with the same constant ''C'' and the same exponent α. The largest such extension is: ::f^*(x):=\inf_\left\. *The image of any U \subset \mathbb^n under an α–Hölder function has Hausdorff dimension at most \tfrac, where \dim_H(U) is the Hausdorff dimension of U . *The space C^(\Omega), 0<\alpha\leq 1 is not separable. *The embedding C^(\Omega)\subset C^(\Omega), 0<\alpha<\beta\leq 1 is not dense. * If f(t) and g(t) satisfy on smooth arc ''L'' the H(\mu) and H(\nu) conditions respectively, then the functions f(t) + g(t) and f(t).g(t) satisfy the H(\alpha) condition on ''L'', where \alpha is the smaller of the numbers \mu, \nu.


Notes


References

* *. * {{DEFAULTSORT:Holder Condition Functional analysis Lipschitz maps Function spaces