Difference between revisions of "User:Camillo.delellis/sandbox"
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==Functions of one variable== | ==Functions of one variable== | ||
| − | ===Definition=== | + | ===Definition 1=== |
| + | Let $I\subset \mathbb R$ be an interval. A function $f: I\to \mathbb R$ is said to have bounded variation if | ||
| + | its [[Variation of a function|total variation]] is bounded. The total variation is defined in the following way. | ||
| + | |||
| + | '''Definition 1''' | ||
| + | Let $I\subset \mathbb R$ be an interval and consider the collection $\Pi$ of ordered $2N$-ples points $a_1<b_1<a_2< b_2 < \ldots < a_N<b_N\in I$, | ||
| + | where $N$ is an arbitrary natural number. The total variation of a function $f: I\to \mathbb R$ is given by | ||
| + | \begin{equation}\label{e:TV} | ||
| + | TV\, (f) := \sup \left\{ \sum_{i=1}^N |f(b_i)-f(a_i)| : (a_1, \ldots, b_N)\in\Pi\right\}\, | ||
| + | \end{equation} | ||
| + | (cp. with Definition ?? of {{cite|AFP}} or Definition ?? of {{Cite|Ro}}). | ||
| + | |||
====Generalizations==== | ====Generalizations==== | ||
| + | The definition of total variation of a function of one real variable can be easily generalized when the target is a [[Metric space|metric space]] $(X,d)$: it suffices to substitute $|f(b_i)-f(a_i)|$ with $d (f(a_i), f(b_i))$ in \ref{e:TV}. Consequently, one defines functions of bounded variation taking values in an arbitrary metric space. Observe that, if $f:I\to X$ is a function of bounded variation and $\varphi:X\to Y$ a [[Lipschitz condition|Lipschitz map]], then $\varphi\circ f$ is also a function of bounded variation and | ||
| + | \[ | ||
| + | TV\, (\varphi\circ f) \leq {\rm Lip (\varphi)}\, TV\, (f)\, , | ||
| + | \] | ||
| + | where ${\rm Lip}\, (\varphi)$ denotes the Lipschitz constant of $\varphi$. | ||
| + | |||
| + | As a corollary we derive | ||
| + | |||
| + | '''Proposition 2''' | ||
| + | A function $(f^1, \ldots, f^k) = f: I\to \mathbb R^k$ is of bounded variation if and only if each coordinate function $f^j$ is of bounded variation. | ||
| + | |||
===General properties=== | ===General properties=== | ||
===Structure theorem=== | ===Structure theorem=== | ||
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===Examples=== | ===Examples=== | ||
===Historical remark=== | ===Historical remark=== | ||
| − | + | Functions of bounded variation were introduced for the first time by C. Jordan in {{Cite|Jo}} to study the pointwise convergence of Fourier series. In particular Jordan proved the following generalization of [[Dirichlet theorem]] on the convergence of Fourier series, called [[Jordan criterion]] | |
| + | |||
| + | '''Theorem''' | ||
| + | Let $f: \mathbb R\to\mathbb R$ be a $2\pi$ periodic square summable function. | ||
| + | * If $f$ has bounded variation in an open interval $I$ then its Fourier series converges to $\frac{1}{2} (f (x^+) + f(x^-))$ at every $x\in I$. | ||
| + | * If in addition $f$ is continuous in $I$ then its Fourier series converges uniformly to $f$ on every closed interval $J\subset I$. | ||
| + | |||
| + | See {{Cite|Zy}} for a proof. | ||
| + | |||
==Functions of several variables== | ==Functions of several variables== | ||
===Historical remarks=== | ===Historical remarks=== | ||
| Line 38: | Line 68: | ||
===Cahn-Hilliard=== | ===Cahn-Hilliard=== | ||
==References== | ==References== | ||
| + | {| | ||
| + | |- | ||
| + | |valign="top"|{{Ref|AFP}}|| L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. {{MR|1857292}}{{ZBL|0957.49001}} | ||
| + | |- | ||
| + | |valign="top"|{{Ref|EG}}|| L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. {{MR|1158660}} {{ZBL|0804.2800}} | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Ha}}|| P.R. Halmos, "Measure theory" , v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}} | ||
| + | |- | ||
| + | |valign="top"|{{Ref|HS}}|| E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) {{MR|0188387}} {{ZBL|0137.03202}} | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Jo}}|| C. Jordan, "Sur la série de Fourier" ''C.R. Acad. Sci. Paris'' , '''92''' (1881) pp. 228–230 | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Ro}}|| H.L. Royden, "Real analysis" , Macmillan (1969) {{MR|0151555}} {{ZBL|0197.03501}} | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Zy}}|| A. Zygmund, "Trigonometric series" , '''1–2''' , Cambridge Univ. Press (1988) | ||
| + | |- | ||
| + | |} | ||
Revision as of 08:48, 19 August 2012
2020 Mathematics Subject Classification: Primary: 26A45 [MSN][ZBL] (Functions of one variable)
2020 Mathematics Subject Classification: Primary: 26B30 Secondary: 28A1526B1549Q15 [MSN][ZBL] (Functions of severable variables)
Functions of one variable
Definition 1
Let $I\subset \mathbb R$ be an interval. A function $f: I\to \mathbb R$ is said to have bounded variation if its total variation is bounded. The total variation is defined in the following way.
Definition 1 Let $I\subset \mathbb R$ be an interval and consider the collection $\Pi$ of ordered $2N$-ples points $a_1<b_1<a_2< b_2 < \ldots < a_N<b_N\in I$, where $N$ is an arbitrary natural number. The total variation of a function $f: I\to \mathbb R$ is given by \begin{equation}\label{e:TV} TV\, (f) := \sup \left\{ \sum_{i=1}^N |f(b_i)-f(a_i)| : (a_1, \ldots, b_N)\in\Pi\right\}\, \end{equation} (cp. with Definition ?? of [AFP] or Definition ?? of [Ro]).
Generalizations
The definition of total variation of a function of one real variable can be easily generalized when the target is a metric space $(X,d)$: it suffices to substitute $|f(b_i)-f(a_i)|$ with $d (f(a_i), f(b_i))$ in \ref{e:TV}. Consequently, one defines functions of bounded variation taking values in an arbitrary metric space. Observe that, if $f:I\to X$ is a function of bounded variation and $\varphi:X\to Y$ a Lipschitz map, then $\varphi\circ f$ is also a function of bounded variation and \[ TV\, (\varphi\circ f) \leq {\rm Lip (\varphi)}\, TV\, (f)\, , \] where ${\rm Lip}\, (\varphi)$ denotes the Lipschitz constant of $\varphi$.
As a corollary we derive
Proposition 2 A function $(f^1, \ldots, f^k) = f: I\to \mathbb R^k$ is of bounded variation if and only if each coordinate function $f^j$ is of bounded variation.
General properties
Structure theorem
Lebesgue decomposition
Examples
Historical remark
Functions of bounded variation were introduced for the first time by C. Jordan in [Jo] to study the pointwise convergence of Fourier series. In particular Jordan proved the following generalization of Dirichlet theorem on the convergence of Fourier series, called Jordan criterion
Theorem Let $f: \mathbb R\to\mathbb R$ be a $2\pi$ periodic square summable function.
- If $f$ has bounded variation in an open interval $I$ then its Fourier series converges to $\frac{1}{2} (f (x^+) + f(x^-))$ at every $x\in I$.
- If in addition $f$ is continuous in $I$ then its Fourier series converges uniformly to $f$ on every closed interval $J\subset I$.
See [Zy] for a proof.
Functions of several variables
Historical remarks
Definition
Consistency in one variable
Functional properties
Structure theorem
Slicing
Tonelli variation
Caccioppoli sets
Reduced boundary
Gauss-Green theorem
Coarea formula
Volpert chain rule
Alberti's rank-one theorem
Functions of special bounded variation
Notable applications
Plateau's problem
Isoperimetry
Hyperbolic conservation laws
Mumford shah functional
Cahn-Hilliard
References
| [AFP] | L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. MR1857292Zbl 0957.49001 |
| [EG] | L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800 |
| [Ha] | P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Zbl 0040.16802 |
| [HS] | E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) MR0188387 Zbl 0137.03202 |
| [Jo] | C. Jordan, "Sur la série de Fourier" C.R. Acad. Sci. Paris , 92 (1881) pp. 228–230 |
| [Ro] | H.L. Royden, "Real analysis" , Macmillan (1969) MR0151555 Zbl 0197.03501 |
| [Zy] | A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) |
Camillo.delellis/sandbox. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Camillo.delellis/sandbox&oldid=27689