Step function
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 <math>f: \mathbb{R} \rightarrow \mathbb{R}</math> is called a step function if it can be written as
- <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> is the indicator function of <math>A</math>:
- <math>\chi_A(x) =
\left\{
\begin{matrix}
1, & \mathrm{if} \; x \in A \\
0, & \mathrm{otherwise}.
\end{matrix}
\right. </math>
In this definition, the intervals <math>A_i</math> can be assumed to have the following two properties:
- The intervals are disjoint, <math>A_i\cap A_j=\emptyset</math> for <math>i\ne j</math>
- The union of the intervals is the entire real line, <math>\cup_{i=1}^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
- 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 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 "steps".
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 <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>f = \sum\limits_{i=0}^n \alpha_i \chi_{A_i}\,</math> is <math>\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.[1]
- The derivative of a step function is the Dirac delta function
- <math>\delta(x) = \begin{cases} +\infty, & x = 0 \\ 0, & x \ne 0 \end{cases}</math>
See also
References
- ↑ Weir, Alan J. Lebesgue integration and measure. Cambridge University Press, 1973. ISBN 0-521-09751-7. Text "chapter 3" ignored (help)