1. Introduction
As usual, let 
be the positive half-line, 
be the set of all nonnegative integers, and let 
be the set of all positive integers. The first examples of orthogonal wavelets on 
related to the Walsh functions and the corresponding wavelets on the Cantor dyadic group have been constructed in [1]; then, in [2] and [3], a multifractal structure of this wavelets is observed and conditions for wavelets to generate an unconditional basis in 
-spaces for all 
have been found. These investigations are continued in [4-10] where among other subjects the algorithms to construct orthogonal and biorthogonal wavelets associated with the generalized Walsh functions are studied. In the present paper, using the Walsh-Fourier transform, we construct nonstationary dyadic wavelets on 
(cf. [11-13], [14, Ch.8]).
Let us denote by 
the integer part of
. For every
, we set
where
. Then
.
The dyadic addition on 
is defined as follows
.
Further, we introduce the notations
where
. Then the Walsh function 
of order 
is 
(see, e.g., [15]).
The Walsh-Fourier transform of every function 
that belongs to 
is defined by
,
.
and extent to the whole space 
in a standard way. The intervals
, 
are called the dyadic intervals of range
. The dyadic topology on 
is generated by the collection of dyadic intervals. A subset 
of 
which is compact in the dyadic topology will be called W-compact.
For any 
we define 
and 
by the following algorithm:
Step 1. For each 
choose
, and
, 
, such that
(1)
for all 
Step 2. Define the masks
(2)
with the coefficients
so that 
for all 
(cf. [15, Sect. 9.7]).
Step 3. For each 
put
, (3)
so that
. (4)
Step 4. Define 
by the formula
. (5)
Further, let us define subspaces 
and 
in 
as follows
,
for all
.
We say that a polynomial 
satisfies the modified Cohen condition if there exists a W-compact subset 
of 
such that
and
. (6)
Theorem. Suppose that the masks 
satisfy the modified Cohen condition with a subset 
and there exists 
such that
for all
,
. (7)
Then for any 
the following properties hold:
a) 
and
;
b) 
and 
are orthonormal basis in 
and
, respectively;
c)
,
.
Moreover, we have
.
Corollary. The system
is an orthonormal basis in
.
We prove this theorem in the next section. Then using the notion of an adapted multiresolution analysis suggested by Sendov [12], we discuss an application of the nonstationary dyadic wavelets to compression of the Weierstrass function and the Swartz function.
2. Proof of the Theorem
At first we prove the orthonormality of
. In view of
let us show that
,
.
Denote by 
the characteristic function of
. For each 
we define
for 
Since 
and, for all
, 
in some neighbourhood of zero, we obtain from Equation (3)
for all
. (8)
Let
where
,
. Letting
, we have
that yields
. By induction, we obtain
According to Equation (8), by Fatou’s lemma, we have
(9)
Consequently, 
and, in view of Equation (5),
. It is known that if 
is constant on dyadic intervals of range
, then 
(see [16, Sect. 6.2]). Therefore, each function 
is constant on
, 
, which implies
.
In view of Equation (7), there exists 
such that
for all
,
.
Hence, for
,
.
It follows from Equation (6) that for some 
for
.
Since
,
.
We have
.
or, taking into account Equation (3),
, 
for
,
.
Applying the dominated convergence theorem we obtain
which means that 
is an orthonormal system.
Now, let us prove an orthonormality of
. For any 
denote
. Then
. (10)
Since
We have
Then from Equation (10)
,
. (11)
Let us define
.
Denote
. Under the unitarity of the matrices
We can write
Using the inverse Fourier-Walsh transform, we have
or,
.
With Equation (11) it yields 
To conclude the proof it remains to show that
. (12)
Note, that by Equation (7) for any 
there exist 
such that 
and, consequently,
. (13)
For any 
the subspace 
is invariant with respect to the shift
. Actually, an arbitrary 
can be approximated by fractions
, with arbitrary large
. Besides, each 
is invariant with respect to the shifts
. By Equation (4) it is clear that
.
Let
. There exist 
such that 
and hence 
for all
. The continuity of 
implies that
. If
, then approximating 
with 
from
and using the invariance of a norm with respect to the shift, we obtain
.
Denote by 
the set of all 
such that
. By the Weiner’s theorem we can write
, for some measurable
. It is clearly that 
and, in view of Equation (13), we have
. Hence, the Equation (12) holds. The theorem is proved.
3. Numerical Experiments
For any
, let
,
. According to [12] an adapted multiresolution analysis (AMRA) of rank 
in 
is a collection of closed subspaces
, 
, which satisfies the following conditions:
1) 
for all
;
2)
;
3) for every 
there is a function 
in 
with a finite support 
such that
is an orthonormal basis of
;
4) for every 
there exists a filter
such that
,
. (14)
The sequence 
from condition (4) is called a scaling sequence for given an AMRA. The corresponding a wavelet sequence 
can be defined by
. (15)
Denote by 
the orthogonal complement of 
in
. It is known that, under some conditions, the system 
is an orthonormal basis of
(for more details, see, e.g., [14, Sect. 8.1]). Moreover, if 
denotes the projection of a function 
on the subset
, then
and
. (16)
Let us denote
and
.
For a given array
the direct non-stationary discrete wavelet transform
maps it into
and
.
The inverse transform is defined as follows
and reconstructs 
by 
and
. According to [12] in order to choose the filter 
to maximize 
in Equation (16), we must solve the following problem.
Problem 1. Let 
be the subset of the 2n-dimensional Euclidean space
, which consists of the points 
satisfying the conditions
. (17)
for
. Find a point 
for which
, (18)
where 
is a 
symmetric matrix.
Problem 1 has a solution since 
is a compact. But, as noted in [12], the numerical solution of this problem is not trivial even for
.
Concerning the standard Haar and Daubechies (with 4 coefficients) discrete transforms see, e.g., [17]; we will denote them as SWTH and SWTD, respectively. We write NSWTH for the simplest case of a multiresolution analysis of rank 1 which is considered in [12, Sect. 3] (see also [13]). The nonstationary Daubechies discrete wavelet transform which corresponds an AMRA of rank 
are defined in [12] and we will use the symbol NSWTDN to denote this transform (see NSWTD1 and NSWTD2 in the tables below).
Method A associated with one of the mentioned above discrete wavelet transforms (cf. [17, Chap.7]) consists of the following steps:
Step 1. Apply the discrete wavelet transform 
times to an input array 
and get the sequence
.
Step 2. Allocate a certain percentage of the wavelet coefficients with lagest absolute value (we choose 10%) and nullify the remaining coefficients.
Step 3. Apply the inverse wavelet transform to the modified arrays of the wavelet coefficients.
Step 4. Calculate
, where 
is a reconstructed array.
In Method B the second step is replaced on the uniform quantization and the forth step is replaced on the calculation of the entropy of a vector, obtained in the third step.
We recall that 
is a vector uniform quantization for given vector
, if
where 
is the length of the quantization interval.
The value 
will be calculated by
The Shannon entropy of 
is defined by the formula
where 
is frequency of the value
.
Let us consider a similar approach associated with the following problem:
Problem 2. Let
. Denote by 
the set of all points 
such that
For every 
we define
for
. Find a point 
for which
(19)
where 
is a 
symmetric matrix.
Given an array
, we define the matrix 
in Problem 1 and Problem 2 by
and
respectively. Here 
for
. Suppose that 
is a solution of Equation (19). Then the direct and inverse nonstationary discrete dyadic wavelet transforms are defined by
, 
,
where 
and
. We
Table 1. Values of the square error corresponding to Method A.
Table 2. Values of the entropy obtained by Method B.
denote these discrete transforms as NSWTL1 if 
and as NSWTL2 if
.
Let us recall that the Weierstrass function is defined as
and the Swartz function is defined as
where
. We will consider arrays 
with elements a8,k =
or a8,k =
,
. Then we use the Matlab function fminsearch to solve the optimization problems in Equations (18) and (19). The results of these numerical experiments are presented in Tables 1 and 2. We see that in several cases the introduced nonstationary dyadic wavelets have an advantage over the classical Haar and Daubechies wavelets.