Step function

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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

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

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 i {\displaystyle \chi _{A_{i}}} {\displaystyle \chi _{A_{i}}} is the indicator function of A i {\displaystyle A_{i}} {\displaystyle A_{i}}:

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

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.

Examples

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, A 0 = R . {\displaystyle A_{0}=\mathbb {R} .} {\displaystyle A_{0}=\mathbb {R} .}
  • The sign function sgn(x), 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 equivalent to the sign function, up to a shift and scale of range ( H = ( sgn + 1 ) / 2 {\displaystyle H=(\operatorname {sgn} +1)/2} {\displaystyle H=(\operatorname {sgn} +1)/2}). 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

  • 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 also define step functions with an infinite number of intervals.[6]

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 A i , {\displaystyle A_{i},} {\displaystyle A_{i},} for i = 0 , 1 , … , n {\displaystyle i=0,1,\dots ,n} {\displaystyle i=0,1,\dots ,n} in the above definition of the step function are disjoint and their union is the real line, then f ( x ) = α i {\displaystyle f(x)=\alpha _{i}} {\displaystyle f(x)=\alpha _{i}} for all x ∈ A i . {\displaystyle x\in A_{i}.} {\displaystyle x\in A_{i}.}
  • The definite integral of a step function is a piecewise linear function.
  • The Lebesgue integral of a step function f = ∑ i = 0 n α i χ A i {\displaystyle \textstyle f=\sum _{i=0}^{n}\alpha _{i}\chi _{A_{i}}} {\displaystyle \textstyle f=\sum _{i=0}^{n}\alpha _{i}\chi _{A_{i}}} is ∫ f d x = ∑ i = 0 n α i ℓ ( A i ) , {\displaystyle \textstyle \int f\,dx=\sum _{i=0}^{n}\alpha _{i}\ell (A_{i}),} {\displaystyle \textstyle \int f\,dx=\sum _{i=0}^{n}\alpha _{i}\ell (A_{i}),} where ℓ ( A ) {\displaystyle \ell (A)} {\displaystyle \ell (A)} is the length of the interval A {\displaystyle A} {\displaystyle A}, and it is assumed here that all intervals A i {\displaystyle A_{i}} {\displaystyle A_{i}} have finite length. In fact, this equality (viewed as a definition) can be the first step in constructing the Lebesgue integral.[7]
  • A discrete random variable is sometimes defined as a random variable whose cumulative distribution function is piecewise constant.[8] In this case, it is locally a step function (globally, it may have an infinite number of steps). Usually however, any random variable with only countably many possible values is called a discrete random variable, in this case their cumulative distribution function is not necessarily locally a step function, as infinitely many intervals can accumulate in a finite region.

See also

References

  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. 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.