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Difference between revisions of "Natural function"

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===Some  characterizations of natural functions===
 
===Some  characterizations of natural functions===
  
Theorem. Let ${E_t}^-=\{s \in E: s \leq t\}$ and ${E_t}^+=\{s \in E: t \leq s\}$. The following conditions are equivalent:
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""Theorem."" Let ${E_t}^-=\{s \in E: s \leq t\}$ and ${E_t}^+=\{s \in E: t \leq s\}$. The following conditions are equivalent:
  
 
(a) $f$ is a natural function;
 
(a) $f$ is a natural function;
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(c) $f$ is a Lipschitz function such that $Lip(f) \leq 1$ and $V(f,E)=sup(E)-inf(E)$.
 
(c) $f$ is a Lipschitz function such that $Lip(f) \leq 1$ and $V(f,E)=sup(E)-inf(E)$.
  
Theorem. Let $X \neq\{0\}$ be a normed space over $\mathbb{R} and let $f:[a,b] \to X$
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""Theorem."" Let $X \neq\{0\}$ be a normed space over $\mathbb{R}$ and let $f:[a,b] \to X$ be a function such that $0<V(f,[a,b])<+/inf$. Then the following conditions are equivalent:
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(a) $f/\alpha$
  
 
A natural function is a Lipschitz function and the smallest Lipschitz constant equals $1$.
 
A natural function is a Lipschitz function and the smallest Lipschitz constant equals $1$.
  
Theorem. A function $f: E \to X$ has bounded variation if and only if there exists a non-decreasing bounded function $\phi : E \to \mathbb{R}$ and a natural function $g: \phi (E) \to X$ such that $f=g\circ\phi$ on $E$.
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""Theorem."" A function $f: E \to X$ has bounded variation if and only if there exists a non-decreasing bounded function $\phi : E \to \mathbb{R}$ and a natural function $g: \phi (E) \to X$ such that $f=g\circ\phi$ on $E$.
  
 
Let $f:[a,b] \to \mathbb{R}$. Then $f$ is a natural function if and only if $f$ is absolutely continuous and $|f'(x)|=1$ a.e. on $[a,b]$.
 
Let $f:[a,b] \to \mathbb{R}$. Then $f$ is a natural function if and only if $f$ is absolutely continuous and $|f'(x)|=1$ a.e. on $[a,b]$.

Revision as of 16:46, 21 April 2016


A function $g: E \to X$ is natural if $V(g,E_a^b)=b-a$ for all $a,b \in E$, $a \leq b $, where $E\subset \mathbb{R}$ is a non-empty bounded set, $E_a^b=\{s \in E: a \leq s \leq b \}$ for $a,b \in E$ ($a \leq b$), $X$ is a metric space with a metric $d$, $V(g,E_a^b)$ variation of $g$ on $E_a^b$.

Some characterizations of natural functions

""Theorem."" Let ${E_t}^-=\{s \in E: s \leq t\}$ and ${E_t}^+=\{s \in E: t \leq s\}$. The following conditions are equivalent:

(a) $f$ is a natural function;

(b) $V(f,{E_x}^-)=x+c$, $x \in E$, where $c=-inf(E)$;

(c) $f$ is a Lipschitz function such that $Lip(f) \leq 1$ and $V(f,E)=sup(E)-inf(E)$.

""Theorem."" Let $X \neq\{0\}$ be a normed space over $\mathbb{R}$ and let $f:[a,b] \to X$ be a function such that $0<V(f,[a,b])<+/inf$. Then the following conditions are equivalent:

(a) $f/\alpha$

A natural function is a Lipschitz function and the smallest Lipschitz constant equals $1$.

""Theorem."" A function $f: E \to X$ has bounded variation if and only if there exists a non-decreasing bounded function $\phi : E \to \mathbb{R}$ and a natural function $g: \phi (E) \to X$ such that $f=g\circ\phi$ on $E$.

Let $f:[a,b] \to \mathbb{R}$. Then $f$ is a natural function if and only if $f$ is absolutely continuous and $|f'(x)|=1$ a.e. on $[a,b]$.

References

[1] V.V. Chistyakov, On the theory of set-valued maps of bounded variation of one real variable, Sbornik: Mathematics 189:5 (1998), 797-819.

[2] M. Małolepszy, On natural functions and Lipschitz functions, Real Analysis Exchange, Vol. 28(1), 2002/2003, 255-264.

How to Cite This Entry:
Natural function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Natural_function&oldid=38607