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{{About|a piecewise constant function|the unit step function|Heaviside step function}} | |||
In [[mathematics]], a [[function (mathematics)|function]] on the [[real number]]s is called a '''step function''' (or '''staircase function''') if it can be written as a [[finite set|finite]] [[linear combination]] of [[indicator function]]s of [[interval (mathematics)|interval]]s. Informally speaking, a step function is a [[piecewise]] [[constant function]] having only finitely many pieces. | |||
[[Image:StepFunctionExample.png|thumb|right|250px|Example of a step function (the red graph). This particular step function is [[Continuous_function#Directional_and_semi-continuity|right-continuous]].]] | |||
==Definition and first consequences== | |||
A function <math>f: \mathbb{R} \rightarrow \mathbb{R}</math> is called a '''step function''' if it can be written as {{Citation needed|date=September 2009}} | |||
:<math>f(x) = \sum\limits_{i=0}^n \alpha_i \chi_{A_i}(x)\,</math> for all real numbers <math>x</math> | |||
where <math>n\ge 0,</math> <math>\alpha_i</math> are real numbers, <math>A_i</math> are intervals, and <math>\chi_A\,</math> (sometimes written as <math>1_A</math>) is the [[indicator function]] of <math>A</math>: | |||
:<math>\chi_A(x) = | |||
\begin{cases} | |||
1 & \mbox{if } x \in A, \\ | |||
0 & \mbox{if } x \notin A. \\ | |||
\end{cases} | |||
</math> | |||
In this definition, the intervals <math>A_i</math> can be assumed to have the following two properties: | |||
# The intervals are [[disjoint set|disjoint]], <math>A_i\cap A_j=\emptyset</math> for <math>i\ne j</math> | |||
# The [[union (set theory)|union]] of the intervals is the entire real line, <math>\cup_{i=0}^n A_i=\mathbb R.</math> | |||
Indeed, if that is not the case to start with, a different set of intervals can be picked for which these assumptions hold. For example, the step function | |||
: <math>f = 4 \chi_{[-5, 1)} + 3 \chi_{(0, 6)}\,</math> | |||
can be written as | |||
: <math>f = 0\chi_{(-\infty, -5)} +4 \chi_{[-5, 0]} +7 \chi_{(0, 1)} + 3 \chi_{[1, 6)}+0\chi_{[6, \infty)}.\,</math> | |||
==Examples== | |||
[[Image:Dirac distribution CDF.svg|325px|thumb|The [[Heaviside step function]] is an often used step function.]] | |||
* A [[constant function]] is a trivial example of a step function. Then there is only one interval, <math>A_0=\mathbb R.</math> | |||
* The [[Heaviside step function|Heaviside function]] ''H''(''x'') is an important step function. It is the mathematical concept behind some test [[Signal (electronics)|signals]], such as those used to determine the [[step response]] of a [[dynamical system (definition)|dynamical system]]. | |||
[[File:Rectangular function.svg|thumb|The [[rectangular function]], the next simplest step function.]] | |||
* The [[rectangular function]], the normalized [[boxcar function]], is the next simplest step function, and is used to model a unit pulse. | |||
=== Non-examples === | |||
* The [[integer part]] function is not a step function according to the definition of this article, since it has an infinite number of intervals. However, some authors define step functions also with an infinite number of intervals.<ref>for example see: {{Cite book | author=Bachman, Narici, Beckenstein | title=Fourier and Wavelet Analysis | publisher=Springer, New York, 2000 | isbn=0-387-98899-8 | chapter =Example 7.2.2}}</ref> | |||
==Properties== | |||
* The sum and product of two step functions is again a step function. The product of a step function with a number is also a step function. As such, the step functions form an [[algebra over a field|algebra]] over the real numbers. | |||
* A step function takes only a finite number of values. If the intervals <math>A_i,</math> <math>i=0, 1, \dots, n,</math> in the above definition of the step function are disjoint and their union is the real line, then <math>f(x)=\alpha_i\,</math> for all <math>x\in A_i.</math> | |||
* The [[Lebesgue integral]] of a step function <math>\textstyle f = \sum\limits_{i=0}^n \alpha_i \chi_{A_i}\,</math> is <math>\textstyle \int \!f\,dx = \sum\limits_{i=0}^n \alpha_i \ell(A_i),\,</math> where <math>\ell(A)</math> is the length of the interval <math>A,</math> and it is assumed here that all intervals <math>A_i</math> have finite length. In fact, this equality (viewed as a definition) can be the first step in constructing the Lebesgue integral.<ref>{{Cite book | author=Weir, Alan J | authorlink= | coauthors= | title=Lebesgue integration and measure | date= | publisher=Cambridge University Press, 1973 | location= | isbn=0-521-09751-7 | unused_data=|chapter= 3}}</ref> | |||
==See also== | |||
*[[Simple function]] | |||
*[[Piecewise defined function]] | |||
*[[Sigmoid function]] | |||
*[[Step detection]] | |||
==References== | |||
{{reflist}} | |||
{{DEFAULTSORT:Step Function}} | |||
[[Category:Special functions]] |
Revision as of 20:00, 31 December 2013
29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church.
In mathematics, a function on the real numbers is called a step function (or staircase function) if it can be written as a finite linear combination of indicator functions of intervals. Informally speaking, a step function is a piecewise constant function having only finitely many pieces.
Definition and first consequences
A function is called a step function if it can be written as Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.
where are real numbers, are intervals, and (sometimes written as ) is the indicator function of :
In this definition, the intervals can be assumed to have the following two properties:
Indeed, if that is not the case to start with, a different set of intervals can be picked for which these assumptions hold. For example, the step function
can be written as
Examples
- A constant function is a trivial example of a step function. Then there is only one interval,
- The Heaviside function H(x) is an important step function. It is the mathematical concept behind some test signals, such as those used to determine the step response of a dynamical system.
- The rectangular function, the normalized boxcar function, is the next simplest step function, and is used to model a unit pulse.
Non-examples
- The integer part function is not a step function according to the definition of this article, since it has an infinite number of intervals. However, some authors define step functions also with an infinite number of intervals.[1]
Properties
- The sum and product of two step functions is again a step function. The product of a step function with a number is also a step function. As such, the step functions form an algebra over the real numbers.
- A step function takes only a finite number of values. If the intervals in the above definition of the step function are disjoint and their union is the real line, then for all
- The Lebesgue integral of a step function is where is the length of the interval and it is assumed here that all intervals have finite length. In fact, this equality (viewed as a definition) can be the first step in constructing the Lebesgue integral.[2]
See also
References
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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534