Shifting operator (E)

Shifting operator (E):

Let y = f (x) be a given function of x and x₀ , x₀+h , x₀+2h , x₀+3h ,…., x₀+nh be the consecutive values of x. Then the operator E is defined as    

E[f(x₀)] = f(x₀+h)

E is called the shifting operator . It is also called the displacement operator.

E[ f ( x ₀ + h)] = f (x ₀ + 2h), E[ f (x₀ + 2h)] = f (x₀ + 3h),...,

E[ f (x₀ + (n −1)h)] = f (x₀ + nh)

E[ f (x)] = f (x + h) , h is the (equal) interval of spacing

E ² f (x) means that the operator E is applied twice on f (x)

(i.e) E²f (x) = E[E f (x)] = E [ f (x + h ) = f (x + 2h)

In general ,

Eⁿ f (x) = f (x + nh) and E^−n f (x) = f (x − nh)

Properties of the operator E:

1. E[f₁(x)+ f₂(x)+…. F_n(x)] = E f₁(x) + E f₂(x)…+…..+E[ f_n(x)]

2. E[cf (x)] =cE[f(x)] c is constant

3. E ^m [ Eⁿ f (x) ] = Eⁿ [ ( E^m f (x) ) = E^{m +n} f ( x)

4. If ‘n’ is a positive integer, then Eⁿ [ E^−n ( f (x)) ] = f (x)

Note

Let y = f(x) be given function of x. Let y₀, y₁, y₂,…. y_n be the values of y at  x = x₀, x₁, x₂,….. x_n.. Then E can also be defined as

Ey₀ = y₁, Ey₁=y₂,….,Ey_n-1=y_n

E[ Ey₀] = E(y₁) = y₂ and in general Eⁿy₀₌y_n