Wavelet Methods - Computational Methods for Quantitative Finance - page 161

approximation order p . In addition, we associated with Φ a dual basis, Φ ={ b ,j :
1

b ,j ,b ,j = δ j,j ,1

≤ j,j ≤ N . The approximation

≤ j ≤ N } , i.e. one has

Φ is denoted by

order of p and we assume p ≤ p .

Example 12.1.1 (Piecewise linear wavelets) We define the wavelet functions ψ , k as
the following piecewise linear functions. Let h = a) and c := √ 3 / 2 2 − − 1 (b

− ·

( 2 h ) − 1 / 2 .For = 0wehave N 0 = 1 and ψ 0 , 1 is the function with value 2 c 0 at

x = a + h 0 .For ≥ 1thewavelet ψ , 1 has the values ψ , 1 (a + h ) = 2 c , ψ , 1 (a +

2 h ) =− c and zero at all other nodes. The wavelet ψ , 2 has the values ψ , 2 (b −

h ) c and zero at all other nodes. The wavelet ψ ,k with

1 <k< 2 has the values ψ ,k (a

= 2 c , ψ , 2 (b − 2 h ) =−

+ ( 2 k − 2 )h ) =− c , ψ ,k (a + ( 2 k − 1 )h ) = 2 c ,

ψ ,k (a 0 ,..., 3, these wavelets

are plotted in Fig. 12.1 . The constants c are chosen such that the wavelets ψ ,k are
normalized in L 2 (G) . Note that these biorthogonal wavelets W are not orthogonal
on V − 1 . But the inner wavelets ψ ,k with 1 <k< 2 have two vanishing moments,

+ 2 kh ) =− c and zero at all other nodes. For =

i.e. ψ ,k (x)x n d x =

0for n = 0 , 1. The approximation order of V is p = 2.

Since V L = span { ψ ,k : 0 ≤ ≤ L, k ∈∇ } , we have a unique decomposition

L L

u = u = u ,k ψ ,k ,

= 0 = 0 k ∈∇

∈ H s (G) ,0

for any u ∈ V L with u ∈ W . Furthermore, any u ≤ s ≤ p admits a

representation as an infinite wavelet series,

∞ ∞

u = u = u ,k ψ ,k , (12.1)

= 0 = 0 k ∈∇

which converges in H s (G) . The coefficients u ,k are the so-called wavelet coeffi-
cients of the function u .

12.1.1 Wavelet Transformation

For u ∈ V L we want to show how to obtain the multi-scale wavelet coefficients

= L = 0 k ∈∇ d ,k ψ ,k , from the single-scale coefficients c ,j in

d ,k := u ,k in u

= N L
j =

u

1 c ,j b ,j .
For the wavelets ψ ,k as in Example 12.1.1 and hat functions b ,j ,wehave

1 <k< 2 ,

ψ ,k =− 0 . 5 b , 2 k − 2 + b , 2 k − 1 − 0 . 5 b , 2 k ,

ψ , 1 = b , 1 − 0 . 5 b , 2 ,

ψ , 2 =− 0 . 5 b , 2 + 1 + b , 2 + 1

1 , − 2 −