Math 2.0

Difference and functional equations Definition

Higher-order differences

Builds on Forward difference operatorBackward difference operator

Definition

Let GG and HH be additive abelian groups, let h∈Gh \in G, and let f ⁣:G→Hf \colon G \to H be a function. Let Δh\Delta_h be the forward difference operator with step hh. The forward differences of order nn of ff, for n∈ℕn \in \N, are defined by induction: Δh0f=f,Δhn+1f=Δh(Δhnf).\Delta_h^0 f = f, \qquad \Delta_h^{n+1} f = \Delta_h\left(\Delta_h^n f\right). The backward differences ∇hnf\nabla_h^n f of order nn are defined in the same way from the backward difference operator ∇h\nabla_h.

Remarks

Example: for f(x)=x2f(x) = x^2 on ℝ\R, Δhf(x)=2hx+h2\Delta_h f(x) = 2hx + h^2 and Δh2f(x)=2h(x+h)+h2−2hx−h2=2h2\Delta_h^2 f(x) = 2h(x + h) + h^2 - 2hx - h^2 = 2h^2. Non-example: (Δhf)2\left(\Delta_h f\right)^2, the square of the first difference, is not Δh2f\Delta_h^2 f; for f(x)=xf(x) = x and h=1h = 1 it is 11, while Δ2f=0\Delta^2 f = 0. In terms of values, Δh2f(x)=f(x+2h)−2f(x+h)+f(x)\Delta_h^2 f(x) = f(x + 2h) - 2f(x + h) + f(x). The two kinds are related by ∇hnf(x)=Δhnf(x−nh)\nabla_h^n f(x) = \Delta_h^n f(x - nh).

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  15. Backward difference operator definition
  16. Forward difference operator definition
  17. Higher-order differences definition

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6 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Finite_difference
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MSC39A70, 39A12
DefinesHigher-order differences
UsesForward difference operator, Backward difference operator

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