The difference operator can be thought of as the discrete version of the derivative operator sending a function to its derivative (if it exists). With the derivative operation, there corresponds an inverse operation called the antiderivative, which, given a function , finds its antiderivative so that the derivative of gives . There is also a discrete analog of this inverse operation, and it is called the indefinite sum.
The indefinite sum of a function is the set of functions
This set is often denoted by or , and any element in is called an indefinite sum of .
Remark. Like the indefinite integral, the indefinite sum is shift invariant. This means that for any , then for any . But, unlike the indefinite integral, the indefinite sum is also invariant by a shift of a periodic real function of period . Conversely, the difference of two indefinite sums of a function is a periodic real function of period .
In the following discussion, we consider the indefinite sum of a function as a function.
, and modulo a real function of period .
Modulo a real number, and treating as an operator taking a function into a function, we see that is linear, that is,
for any , and
If , then .
Below is a table of some basic functions and their indefinite sums ( is a real-valued periodic function with period ):