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Step function

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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.

Contents

Definition and first consequences

A function f : R R is called a step function if it can be written as

f ( x ) = i = 0 n α i χ A i ( x ) for all real numbers x

where n 0 , α i are real numbers, A i are intervals, and χ A (sometimes written as 1 A ) is the indicator function of A :

χ A ( x ) = { 1 if  x A , 0 if  x A .

In this definition, the intervals A i can be assumed to have the following two properties:

  1. The intervals are pairwise disjoint, A i A j   =   for i     j
  2. The union of the intervals is the entire real line, i = 0 n A i   =   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 )

can be written as

f = 0 χ ( , 5 ) + 4 χ [ 5 , 0 ] + 7 χ ( 0 , 1 ) + 3 χ [ 1 , 6 ) + 0 χ [ 6 , ) .

Examples

  • A constant function is a trivial example of a step function. Then there is only one interval, A 0 = 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 an important step function, and is equivalent to the sign function, up to a shift and scale of range ( H = ( 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 normalized boxcar function, is the next simplest step function, and is used to model a unit pulse.
  • 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.
  • 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 , i = 0 , 1 , , n , in the above definition of the step function are disjoint and their union is the real line, then f ( x ) = α i for all x 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 is f d x = i = 0 n α i ( A i ) , where ( A ) is the length of the interval A , and it is assumed here that all intervals A i have finite length. In fact, this equality (viewed as a definition) can be the first step in constructing the Lebesgue integral.
  • References

    Step function Wikipedia