DISCRETE WAVELET TRANSFORMS IN WALSH ANALYSIS

We choose a number α ∈ (0, 1) and consider the case where the components of the vector b are
given by the equalities
{
α, if k =0ork = N − 1,
b k =
1, otherwise.
Then for Φ a N
(x), the following equality holds:
NΦ a N (x) =D N (x) − (1 − α)(1 + w N−1 (x)),
where
D N (x) :=
N−1
∑
j=0
w j (x) =
{
N, x ∈ [0,N −1 ),
0, x ∈ [N −1 , 1).
The last equality expresses the well-known property of Dirichlet–Walsh-type kernels (see, e.g., [25,
Sec. 1.5]).
For any functions f n ∈ V n and g n ∈ W n , the following equalities are valid:
where
f n (x) =
N−1
∑
k=0
g (s)
n
p−1
∑
C n,k ϕ n,k (x), g n (x) =
N−1
(x) = ∑
In these formulas, the sequences of coefficients
C n = {C n,k },
D (s)
n
k=0
D (s)
n,k ψ(s) n,k (x).
s=0
g n
(s) (x),
= {D (s)
n,k
}, 1 ≤ s ≤ p − 1,
are uniquely determined by f n and g n , respectively. As was shown in [13], an arbitrary function
f n+1 ∈ V n+1 can be decomposed into the direct sum of the functions f n ∈ V n and g n
(s) ∈ W n
(s) ,and
also f n+1 can be reconstructed from this decomposition by using the following discrete transforms.
1. The direct periodic discrete wavelet transform
∑p−1
N−1
∑
C n,ν = A (n)
pk+j,ν C n+1,pk+j, D (s)
where
j=0 k=0
p−1
n,ν = ∑
j=0
B (n)
pν+j,s C n+1,pν+j, 0 ≤ ν ≤ N − 1, (14)
1
⎧⎪ ⎨
A (n)
pk+j,ν = p + 1 − α
αpN , ν = k,
B (n)
⎪ ⎩ ε γ(ν)−γ(k) 1 − α
p
αpN , ν ≠ k, pk+j,s = p−1 ε −js
p .
Recall that we denote by γ(k) the sum of digits in p-ary decomposition of the number k.
2. The inverse periodic discrete wavelet transform
where
134
C n+1,pk+j =
N−1
∑
ν=0
Q (n)
pk+j,ν C ∑
n,ν +
Q (n)
⎧
⎪⎨ 1 − ε γ(ν)
pk+j,ν = p
⎪ ⎩ − ε γ(ν)
p
p−1
ε js
p D (s)
n,k
s=1
, 0 ≤ k ≤ N − 1, 0 ≤ j ≤ p − 1, (15)
(1 − α) γ n+1,pk+j
, k = ν,
N
(1 − α) γ n+1,pk+j
, k ≠ ν.
N