One Dimension

We follow Chapter 13 of Carothers [#!kn:Carothers2000!#], especially pages 202 to 203 and Chapter 2 of Niederreiter [#!kn:Niederreiter1992!#]. In this section we use Carothers terminology and notation.

We start with a function $f$ of a single variable over a finite closed interval from a to b. Use the variable t as the independent variable, so t varies from a to b inclusive, $[a,b]$.

Definition 5 (Partition P of [a,b)   ] A set, P, of points is a partition of the set is specified as follows. P is a set of points $t_i$,i=0,...,n such that $t_i < t_{i+1}$, $t_0=a$ and $t_0=b$.

Definition 6 (Refinement Q of a Partition)   A partition Q is a refinement of P, if every point of P is a point of Q.

Definition 7 (Variation for a Partition)  

$$V(f,P) = \sum_{i=1}^n \vert f(t_i)-f(t_{i-1})\vert$$ (2.2)

Definition 8 (Total Variation)   The total variation of f over [a,b] is defined by

$$V_{a}^{b}f = sup_{P} V(f,P)$$ (2.3)

Here sup is the supremum over all possible partitions of the interval [a,b].