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Discrete Calculus: Linearity


July 27, 2020 · 23:42 pm · Discrete CalculusMath

For an operator to be a linear function, they must meet two criterias:

  1. Additivity, i.e. addition is preserved
  2. Homogeneity of degree 1, i.e. scalar multiplication is preserved

In other words, the following are satisfied:

  • f(x+y)=f(x)+f(y)f(x+y) = f(x) + f(y)
  • f(cx)=cf(x)f(cx) = c f(x)

for the function ff of interest. It turns out that discrete derivatives satify these conditions.

Additivity

Δh[f+g](n)=[f+g](n+h)[f+g](n)h=f(n+h)+g(n+h)(f(n)+g(n))h=f(n+h)f(n)h+g(n+h)g(n)h=Δhf(n)+Δhg(n) \begin{aligned} \Delta_h[f + g](n) &= \frac{[f+g](n + h) - [f+g](n)}{h} \\ &= \frac{f(n + h) + g(n + h) - (f(n) + g(n))}{h} \\ &= \frac{f(n+h)-f(n)}{h} + \frac{g(n+h)-g(n)}{h} \\ &= \Delta_h f(n) + \Delta_h g(n) \space \blacksquare \\ \end{aligned}

Homogeneity of degree 1

Δh[cf](n)=[cf](n+h)[cf](n)h=cf(n+h)cf(n)h=cf(n+h)f(n)h=cΔhf(n) \begin{aligned} \Delta_h[cf](n) &= \frac{[cf](n + h) - [cf](n)}{h} \\ &= \frac{c f(n+h) - c f(n)}{h} \\ &= c \cdot \frac{f(n+h) - f(n)}{h} \\ &= c \Delta_h f(n) \space \blacksquare \\ \end{aligned}

So we can conclude that the discrete derivative as an operation is a linear transformation in the function space.


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