Relations between the operators Δ, ∇ and E

Relations between the operators Δ, ∇ and E:

1. Δ ≡ E −1

Proof:         From the definition of Δ we know that

Δ f (x) = f (x + h ) − f (x) and   

E[ f (x )] = f (x + h)

where h is the interval of difference.

Δ f (x) = f (x + h ) − f (x)

Δ f (x) = Ef (x ) − f (x)

⇒ Δ f (x) = (E −1) f (x)

Δ ≡ E −1

∴    E ≡ 1 + Δ

2.   Δ E       ≡       Δ E

Proof:

E(Δf(x)) = E[f(x+h)- f(x)]

= E f(x+h) - E f(x)

= f(x+2h) - f(x+h)

=Δf(x+h)

=ΔEf(x)

ΔE ≡ ΔE

3. ∇≡ E −1 / E

∇f (x) = f (x) - f (x-h)

= f (x) – E^-1f (x)

= (1- E^-1) f (x)

∇ ≡ (1- E^-1)

∇ ≡ 1 – 1/E

Hence ∇ ≡ [E – 1]/E