Step function (original) (raw)

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Linear combination of indicator functions of real intervals

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 f : R → R {\displaystyle f\colon \mathbb {R} \rightarrow \mathbb {R} } {\displaystyle f\colon \mathbb {R} \rightarrow \mathbb {R} } is called a step function if it can be written as [_citation needed_]

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

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

χ A ( x ) = { 1 if x ∈ A 0 if x ∉ A {\displaystyle \chi _{A}(x)={\begin{cases}1&{\text{if }}x\in A\\0&{\text{if }}x\notin A\\\end{cases}}} {\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 A i {\displaystyle A_{i}} {\displaystyle A_{i}} can be assumed to have the following two properties:

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

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

can be written as

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

  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](/wiki/Template:Cite%5Fweb "Template: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](/wiki/Template:Cite%5Fbook "Template: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.