Math 2.0

Difference and functional equations Definition

Backward difference operator

Builds on Finite differenceFunction

Definition

Let GG and HH be additive abelian groups, and let h∈Gh \in G. The backward difference operator with step hh sends a function f ⁣:G→Hf \colon G \to H to the function ∇hf ⁣:G→H\nabla_h f \colon G \to H given by ∇hf(x)=f(x)−f(x−h)(x∈G).\nabla_h f(x) = f(x) - f(x - h) \qquad (x \in G). When h=1h = 1, one writes ∇\nabla for ∇1\nabla_1. For a sequence u ⁣:ℤ→Hu \colon \Z \to H, this gives (∇u)n=un−un−1(\nabla u)_n = u_n - u_{n-1}.

Remarks

It is a finite difference in the backward direction. Example: for f(x)=x2f(x) = x^2 on ℝ\R, ∇hf(x)=x2−(x−h)2=2hx−h2\nabla_h f(x) = x^2 - (x - h)^2 = 2hx - h^2. Non-example: f(x+h)−f(x)f(x + h) - f(x) is in general not ∇hf(x)\nabla_h f(x); for f(x)=x2f(x) = x^2 it equals 2hx+h22hx + h^2. The two operators are related by ∇hf(x)=Δhf(x−h)\nabla_h f(x) = \Delta_h f(x - h), where Δh\Delta_h is the forward difference operator. In operator form, ∇h=I−E−h\nabla_h = I - E_{-h}, with E−hf(x)=f(x−h)E_{-h} f(x) = f(x - h) a shift operator.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Finite_difference
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