Step function

From Wikipedia, the free encyclopedia
Jump to: navigation, search
This article is about a piecewise constant function. For the unit step function, see Heaviside 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.

Example of a step function (the red graph). This particular step function is right-continuous.

Definition and first consequences[edit]

A function is called a step function if it can be written as[citation needed]

for all real numbers

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:

  1. The intervals are pairwise disjoint, for
  2. The union of the intervals is the entire real line,

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[edit]

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,
  • The sign function , which is −1 for negative numbers and +1 for positive numbers, and is the simplest non-constant step function.
  • The Heaviside function H(x), which is 0 for negative numbers and 1 for positive numbers, is an important step function, and is equivalent to the sign function, up to a shift and scale of range (). 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 next simplest step function.

Non-examples[edit]

  • 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[edit]

  • 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 definite integral of a step function is a piecewise linear function.
  • 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[edit]

References[edit]

  1. ^ for example see: Bachman, Narici, Beckenstein. "Example 7.2.2". Fourier and Wavelet Analysis. Springer, New York, 2000. ISBN 0-387-98899-8. 
  2. ^ Weir, Alan J. "3". Lebesgue integration and measure. Cambridge University Press, 1973. ISBN 0-521-09751-7.