B.2. PIECEWISE LINEAR APPROXIMATION 377

B.2 Piecewise Linear ApproximationDefinition B.2.1 Let P be a dissection of [0,T ] and let F ∈ V p ([0,T ]) . Let FP

denote the piecewise linear approximation of F. That is, it agrees with F at every pointof P and in [ti, ti+1] it is of the form 1

ti+1−ti[(F (ti))(ti+1− t)+F (ti+1)(t− ti)] = F (ti)+

(t− ti)(

F(ti+1)−F(ti)ti+1−ti

).

To get the piecewise linear approximation, you could write

FP (t)≡∫ t

0

(F (0)t1− t0

+n−1

∑i=0

F (ti+1)−F (ti)ti+1− ti

)X[ti,ti+1) (s)ds

Note that the formula gives

FP (t1) = F (0)+F (t1)−F (t0) = F (t1) ,

FP (t2) = F (t1)+F (t2)−F (t1)

t2− t1(t2− t1) = F (t2)

etc.Next is a fundamental approximation lemma which says that when you replace a func-

tion in V p with its piecewise linear approximation the p variation gets smaller. First is asimple observation. Suppose p ≥ 1 and ∑

ni=1 ri = r where each ri is positive and less than

1. Then rp ≥ ∑ni=1 rp

i . To see this is so, note that ∑ni=1

rir = 1 and so ∑i

( rir

)p ≤ ∑irir = 1 so

the claim follows.

Lemma B.2.2 Let F ∈ V p ([0,T ]) and let P be a dissection. Then∥∥FP

∥∥p,[0,T ] ≤

∥F∥p,[0,T ]. Also, if Pε is a dissection for which

∑Pε

∣∣∣FP (ui+1)−FP (ui)∣∣∣p > ∥∥∥FP

∥∥∥p

p,[0,T ]− ε,

then this inequality continues to hold for Pε in which Pε ⊆P .

Proof: Let Pε be a dissection of [0,T ] such that

∑ui∈Pε

∣∣∣FP (ui+1)−FP (ui)∣∣∣p > ∥∥∥FP

∥∥∥p

p,[0,T ]− ε (2.1)

The idea is to show that, deleting some points of Pε you can assume every grid point ofPε is also a grid point of P while preserving the above inequality. This might not be toosurprising because the variation of FP is determined by the F (ti) where ti ∈P . Let themesh points of Pε be denoted by {ui} and those of P are denoted by

{t j}

.Let

(t j, t j+1

)be an interval which contains points of Pε say ui,ui+1, ...,uk in order.

Modify P if necessary by including t j, t j+1. Then corresponding to this interval, the abovesum gives

k−1

∑l=i|a(ul+1−ul)|p +

∣∣a(t j+1−uk)∣∣p + ∣∣a(ui− t j)

∣∣p