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Difference and functional equations Definition

Shift operator

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Definition

Let GG be an additive abelian group, let YY be a set, and let h∈Gh \in G. The shift operator EhE_h with step hh sends a function f ⁣:G→Yf \colon G \to Y to the function Ehf ⁣:G→Y,Ehf(x)=f(x+h).E_h f \colon G \to Y, \qquad E_h f(x) = f(x + h). For a sequence u=(un)n≥0u = (u_n)_{n \ge 0} in YY, the (left) shift EE is defined by (Eu)n=un+1(E u)_n = u_{n+1} for n≥0n \ge 0.

Remarks

Example: for f(x)=sin⁡xf(x) = \sin x and h=π/2h = \pi/2, Ehf(x)=sin⁡(x+π/2)=cos⁡xE_h f(x) = \sin(x + \pi/2) = \cos x. Non-example: f↦f(2x)f \mapsto f(2x) is not a shift operator; it rescales the argument instead of translating it. The shifts compose by EhEk=Eh+kE_h E_k = E_{h+k}, and E0E_0 is the identity. When YY is an abelian group, EhE_h is additive, and the forward difference operator is Δh=Eh−I\Delta_h = E_h - I. Other notations are ThT_h and τh\tau_h; on sequences the left shift is also called the backward shift.

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Further reading

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