Step function

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In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of half-open intervals. Informally speaking, a step function is a piecewise constant function having only finitely many pieces.

Example of a step function with n=4.

Let the following quantities be given:

a sequence of coefficients

$\{\alpha_0, \dots, \alpha_n\}\subset \mathbb{R},\; n \in \mathbb{N} \setminus \{0\}$
a sequence of interval margins

$\{x_1 < \dots < x_{n-1}\} \subset \mathbb{R}$
a sequence of intervals

$A_0 := (-\infty, x_1)$

$A_i := [x_i, x_{i+1})\,$ (for $i=1,\cdots,n-2$ )

$A_n := [x_{n-1},\infty)$

(Although the intervals are shown as being closed below and open above, this is not necessary to the definition; all that is required is that the intervals A_n do not intersect and that their union is the set of real numbers.)

Definition: Given the notations above, a function $f: \mathbb{R} \rightarrow \mathbb{R}$ is a step function if and only if it can be written as

$f(x) = \sum\limits_{i=0}^n \alpha_i \cdot 1_{A_i}(x)$ for all $x \in \mathbb{R}$ where $1_A\,$ is the indicator function of $A\,$ :

$1_A(x) = \left\{ \begin{matrix} 1, & \mathrm{if} \; x \in A \\ 0, & \mathrm{otherwise}. \end{matrix} \right.$

Note: for all $i=0,\cdots,n$ and $x \in A_i$ it holds: $f(x)=\alpha_i\,$ .

A particular step function, the unit step function or Heaviside step function H(x), is obtained by setting n=1, α₀=0, α₁=1, and x₁=0 in the general expression above. It is the mathematical concept behind some test signals, as those used to determine the step response of a dynamical system.

Retrieved from "http://en.wikipedia.org/wiki/Step_function"

Category: Elementary special functions

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Step function

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