Ramp function

Definitions

The ramp function (R(x) : ℝ → ℝ) may be defined analytically in several ways. Possible definitions are:

A system of equations:

The max function:

The mean of a straight line with unity gradient and its modulus:

this can be derived by noting the following definition of max(a,b), max ( a , b ) = a + b + | a − b | 2 for which a = x and b = 0

The Heaviside step function multiplied by a straight line with unity gradient:

The convolution of the Heaviside step function with itself:

The integral of the Heaviside step function:

Macaulay brackets:

Non-negativity

In the whole domain the function is non-negative, so its absolute value is itself, i.e.

∀ x ∈ R : R ( x ) ≥ 0

and

| R ( x ) | = R ( x )

Proof: by the mean of definition 2, it is non-negative in the first quarter, and zero in the second; so everywhere it is non-negative.

Derivative

Its derivative is the Heaviside function:

R ′ ( x ) = H ( x ) for  x ≠ 0.

Second derivative

The ramp function satisfies the differential equation:

d 2 d x 2 R ( x − x 0 ) = δ ( x − x 0 ) ,

where δ(x) is the Dirac delta. This means that R(x) is a Green's function for the second derivative operator. Thus, any function, f(x), with an integrable second derivative, f″(x), will satisfy the equation:

f ( x ) = f ( a ) + ( x − a ) f ′ ( a ) + ∫ a b R ( x − s ) f ″ ( s ) d s for  a < x < b .

Fourier transform

F { R ( x ) } ( f ) = ∫ − ∞ ∞ R ( x ) e − 2 π i f x d x = i δ ′ ( f ) 4 π − 1 4 π 2 f 2 ,

where δ(x) is the Dirac delta (in this formula, its derivative appears).

Laplace transform

The single-sided Laplace transform of R(x) is given as follows,

L { R ( x ) } ( s ) = ∫ 0 ∞ e − s x R ( x ) d x = 1 s 2 .

Iteration invariance

Every iterated function of the ramp mapping is itself, as

R ( R ( x ) ) = R ( x ) .

Proof:

This applies the non-negative property.