DISCRETE WAVELET TRANSFORMS IN WALSH ANALYSIS

and introduce the sequence of filters
with the coefficients
p j+1 −1
j+1 (ω) := ∑
c (0)
j+1,k w k(ω), j ∈ Z + , ω ∈ R + , (9)
m (0)
k=0
p j+1
c (0)
j+1,k = ∑−1
p−j−1
l=0
b (0)
j+1,l w k((j +1)/l). (10)
Applying the inversion formula of the discrete Vilenkin—Krestenson transform (see [25, Sec. 11]), from
(9) and (10) we obtain
b (0)
j+1,k = m(0) j+1 ((j +1)/k), k =0, 1,...,pj+1 − 1, j ∈ Z + .
Next, we define the unitary matrices
⎛
b (0)
j+1,νp
b (0)
j+1,νp+1
... b (0) ⎞
j+1,νp+(p−1)
M ν,j =
b (1)
j+1,νp
b (1)
j+1,νp+1
... b (1)
j+1,νp+(p−1)
⎜
⎝.....................................
⎟
⎠ ,
b (p−1)
j+1,νp
b (p−1)
j+1,νp+1
... b (p−1)
j+1,νp+(p−1)
where ν =0, 1,...,p j − 1. The first rows in M ν,j are formed by numbers from (8), and the remaining
row are defined as in [34, Sec. 2.6] or [14, p. 24]. Now, just like in (10), we set
p j+1
c (s)
j+1,k = ∑−1
p−j−1
l=0
b (s)
j+1,l w k((j +1)/l), s =1,...,p− 1,
and define the nonstationary discrete wavelet transform:
a j,k = ∑
j+1,l⊖pk a j+1,l, d (1)
j,k = ∑
j+1,l⊖pk a j+1,l, ..., d (p−1)
j,k
= ∑
l∈Z +
c (0)
l∈Z +
c (1)
l∈Z +
c (p−1)
j+1,l⊖pk a j+1,l. (11)
Its inversion formula is as follows:
a j+1,l = ∑
c (0)
j+1,l⊖pk a j,k + c (1)
j+1,l⊖pk d(1) j,k
+ ···+ c(p−1)
j+1,l⊖pk d(p−1) j,k
. (12)
l∈Z +
Numerical experiments performed in [21] demonstrate the possibility of using discrete transforms (11)
and (12) for compressing fractal signals. In these experiments, at each approximation level, the transform
coefficients (11) and (12) were chosen so that the projection of the signal onto the corresponding
approximating space was maximum (see [38]).
Some generalizations and modifications of the discrete wavelet transforms discussed above can be
obtained by extending the sets of admissible parameter values as a result of the transition from wavelets
to frames by analogy with the constructions in [2, 6]. In this connection, we note that according to [14,
18], a rigid frame in L 2 (G p ) can be constructed along any vector b =(b 0 ,b 1 ,...,b N−1 ) satisfying the
condition
∣ ∣
b 0 =1, ∣b l 2 ∣ ∣
+ b l+N1 2 ∣ ∣
+ ···+ b l+(p−1)N1 2 ≤ 1, 0 ≤ l ≤ N1 − 1. (13)
In addition, instead of above-mentioned constructions of wavelets and frames on the group G p ,
one can use their generalizations for the Vilenkin group G P associated with the sequence P =
132