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Step function - Wikipedia

  • ️Sat Sep 12 2015

From Wikipedia, the free encyclopedia

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 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.

An example of step functions (the red graph). In this function, each constant subfunction with a function value αi (i = 0, 1, 2, ...) is defined by an interval Ai and intervals are distinguished by points xj (j = 1, 2, ...). This particular step function is right-continuous.

Definition and first consequences

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A function {\displaystyle f\colon \mathbb {R} \rightarrow \mathbb {R} } is called a step function if it can be written as [citation needed]

{\displaystyle f(x)=\sum \limits _{i=0}^{n}\alpha _{i}\chi _{A_{i}}(x)}, for all real numbers {\displaystyle x}

where {\displaystyle n\geq 0}, {\displaystyle \alpha _{i}} are real numbers, {\displaystyle A_{i}} are intervals, and {\displaystyle \chi _{A}} is the indicator function of {\displaystyle A}:

{\displaystyle \chi _{A}(x)={\begin{cases}1&{\text{if }}x\in A\\0&{\text{if }}x\notin A\\\end{cases}}}

In this definition, the intervals {\displaystyle A_{i}} can be assumed to have the following two properties:

  1. The intervals are pairwise disjoint: {\displaystyle A_{i}\cap A_{j}=\emptyset } for {\displaystyle i\neq j}
  2. The union of the intervals is the entire real line: {\displaystyle \bigcup _{i=0}^{n}A_{i}=\mathbb {R} .}

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

{\displaystyle f=4\chi _{[-5,1)}+3\chi _{(0,6)}}

can be written as

{\displaystyle f=0\chi _{(-\infty ,-5)}+4\chi _{[-5,0]}+7\chi _{(0,1)}+3\chi _{[1,6)}+0\chi _{[6,\infty )}.}

Variations in the definition

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Sometimes, the intervals are required to be right-open[1] or allowed to be singleton.[2] The condition that the collection of intervals must be finite is often dropped, especially in school mathematics,[3][4][5] though it must still be locally finite, resulting in the definition of piecewise constant functions.

The Heaviside step function is an often-used step function.
The rectangular function, the next simplest step function.
  • 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[6] also define step functions with an infinite number of intervals.[6]
  1. ^ "Step Function".
  2. ^ "Step Functions - Mathonline".
  3. ^ "Mathwords: Step Function".
  4. ^ "Archived copy". Archived from the original on 2015-09-12. Retrieved 2024-12-16.{{cite web}}: CS1 maint: archived copy as title (link)
  5. ^ "Step Function".
  6. ^ a b Bachman, Narici, Beckenstein (5 April 2002). "Example 7.2.2". Fourier and Wavelet Analysis. Springer, New York, 2000. ISBN 0-387-98899-8.{{cite book}}: CS1 maint: multiple names: authors list (link)
  7. ^ Weir, Alan J (10 May 1973). "3". Lebesgue integration and measure. Cambridge University Press, 1973. ISBN 0-521-09751-7.
  8. ^ Bertsekas, Dimitri P. (2002). Introduction to Probability. Tsitsiklis, John N., Τσιτσικλής, Γιάννης Ν. Belmont, Mass.: Athena Scientific. ISBN 188652940X. OCLC 51441829.