Changes

<math>\operatorname{E}</math> is called a "shift operator" because it takes us from contemplating the value of <math>f\!</math> at a place <math>x\!</math> to considering the value of <math>f\!</math> at a shift of <math>y\!</math> away.  Thus, <math>\operatorname{E}</math> tells us the absolute effect on <math>f\!</math> that is obtained by changing its argument from <math>x\!</math> by an amount that is equal to <math>y\!</math>.

<math>\operatorname{E}</math> is called a "shift operator" because it takes us from contemplating the value of <math>f\!</math> at a place <math>x\!</math> to considering the value of <math>f\!</math> at a shift of <math>y\!</math> away.  Thus, <math>\operatorname{E}</math> tells us the absolute effect on <math>f\!</math> that is obtained by changing its argument from <math>x\!</math> by an amount that is equal to <math>y\!</math>.

'''Historical Note.''' The "shift operator" <math>\operatorname{E}</math> was originally called the "enlargement operator", hence the initial "E" of the usual notation.

Historical Note.  The protean "shift operator" E was originally called

the "enlargement operator", hence the initial "E" of the usual notation.

The next operator, D, associates with a function f : X -> Y

The next operator, <math>\operatorname{D}</math>, associates with a function <math>f : X \to Y</math> another function <math>\operatorname{D}f</math>, where <math>\operatorname{D}f : X \times X \to Y</math> is defined by the following equation:

another function Df, where Df : X x X -> Y is defined by the

following equation:

  Df(x, y) = Ef(x, y) - f(x),

: <math>\operatorname{D}f(x, y) = \operatorname{E}f(x, y) - f(x),</math>

or, equivalently,

or, equivalently,

  Df(x, y) = f(x + y) - f(x).

: <math>\operatorname{D}f(x, y) = f(x + y) - f(x).</math>

D is called a "difference operator" because it tells us about the

<math>\operatorname{D}</math> is called a "difference operator" because it tells us about the relative change in the value of <math>f\!</math> along the shift from <math>x\!</math> to <math>x + y.\!</math>

relative change in the value of f along the shift from x to x + y.

 

In practice, one of the variables, x or y, is often

In practice, one of the variables, x or y, is often

considered to be "less variable" than the other one,

considered to be "less variable" than the other one,