NationMaster - Encyclopedia: Step function

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. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Partial plot of a function f. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... In the mathematical subfield of set theory, the indicator function, or characteristic function, is a function defined on a set X which is used to indicate membership of an element in a subset A of X. Remark. ... In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. ...

Example of a step function with n=4.

Let the following quantities be given: Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ...

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 In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... It has been suggested that this article or section be merged with Logical biconditional. ...

$f(x) = sumlimits_{i=0}^n alpha_i cdot 1_{A_i}(x)$ 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: $f(x)=alpha_i,$ . In the mathematical subfield of set theory, the indicator function, or characteristic function, is a function defined on a set X which is used to indicate membership of an element in a subset A of X. Remark. ...

Special step functions

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. The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of... Look up signal, signaling in Wiktionary, the free dictionary. ... In control theory the unit step response is the response of a dynamic system to the Heaviside step function. ... The dynamical system concept is a mathematical formalization for any fixed rule which describes the time dependence of a points position in its ambient space. ...

Category: Elementary special functions In mathematics, especially in mathematical analysis, a simple function is a measurable function whose range is finite. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...

Results from FactBites:

Heaviside step function - Wikipedia, the free encyclopedia (388 words)
	The function is used in the mathematics of control theory and signal processing to represent a signal that switches on at a specified time and stays switched on indefinitely.
	The Heaviside function is an antiderivative of the Dirac delta function, H' = δ.
	This function is the cumulative summation of the Kronecker delta:

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See also

Special step functions