Step function: Difference between revisions

Revision as of 15:47, 20 May 2005

Revision as of 04:18, 21 May 2005

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AIn [[mathematics]], a [[function (mathematics)|function]] on the reals[[real <math>\Re</math>number|real]]s '''R''' is a '''step function''' if it can be written as a finite [[linear combination]] of [[characteristic function]]s of semi-open intervals.

Let the following quantities be given:

<ul>

<li> a [[sequence]] of coefficients
:<math>(\{\alpha_0, \dots, \alpha_n)\}\subset \Remathbb{R},\; n \in \mathrmmathbb{N} \setminus \{0\}</math> </li>

<li> a sequence of interval margins
:<math>(\{x_1 < \dots < x_{n-1})\} \subset \Remathbb{R}</math> </li>

<li> a [[set]] of functions <math>p_i:\Re\rightarrow\Re, \;\; i \in \overline{0,n}</math> such that: <br>

<li>

a set of functions <math>p_i:\Re\rightarrow\Re, \;\; i \in \overline{0,n}</math> such that: <br>

:<math> p_0(x) = \left\{

\begin{matrix}

1, & \mathrm{if} \; x < x_0 \\

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\end{matrix}

\right.

</math> <br>

:<math> \forall i \in \overline{1, n-1} \;\;

p_i(x) = \left\{

\begin{matrix}

Line 21 ⟶ 23:

\end{matrix}

\right.

</math> <br>

:<math> p_n(x) = \left\{

\begin{matrix}

0, & \mathrm{if} \; x \leq x_{n-1} \\

Line 28 ⟶ 30:

\end{matrix}

\right.

</math> <br>

</li>

</ul>

'''Definition:''' Given the notationnotations above, a function <math>''f'':\Re\rightarrow\Re</math>'''R'''&rarr;'''R''' is a '''step function''' [[if and only if]] it could be written as

:<math>

f(x) = \sum\limits_{i=0}^n \alpha_i \cdot p_i(x).

</math>

'''Note:''' Step functions are piecewise functions constant on each interval <math>[x_{i-1}, x_i).</math>

== Special step functions ==

[[Heaviside step function]]

== Related topictopics ==

*[[piecewisesimple function]]

*[[piecewise function]]

[[ro:functia scara]]

[[Category:special functions]]