Math 2.0

Difference and functional equations Definition

Forward difference operator

Builds on Finite differenceFunction

Definition

Let MM be an additive commutative monoid, let GG be an additive abelian group, and let h∈Mh \in M. The forward difference operator with step hh sends a function f ⁣:M→Gf \colon M \to G to the function Δhf ⁣:M→G\Delta_h f \colon M \to G given by Δhf(x)=f(x+h)−f(x)(x∈M).\Delta_h f(x) = f(x + h) - f(x) \qquad (x \in M). When h=1h = 1, one writes Δ\Delta for Δ1\Delta_1. For a sequence u ⁣:ℕ→Gu \colon \N \to G, this gives (Δu)n=un+1−un(\Delta u)_n = u_{n+1} - u_n.

Remarks

It is a finite difference in the forward direction. The operator Δh\Delta_h is additive: Δh(f+g)=Δhf+Δhg\Delta_h(f + g) = \Delta_h f + \Delta_h g. Example: for f(x)=2xf(x) = 2^x on ℤ\Z with h=1h = 1, Δf(x)=2x+1−2x=2x\Delta f(x) = 2^{x+1} - 2^x = 2^x, so ff is a fixed point of Δ\Delta, like exe^x for the derivative. Non-example: g(x)=f(x+1)−f(x−1)g(x) = f(x+1) - f(x-1) is not Δhf\Delta_h f for any hh; it is a central difference with step 22. With the shift operator Ehf(x)=f(x+h)E_h f(x) = f(x + h) and the identity II, one has Δh=Eh−I\Delta_h = E_h - I.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Finite_difference
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MSC39A70, 39A12
DefinesForward difference operator
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