A new class of WA-systems of Kravchenko-Rvachev functions

Full text

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325

ISSN 1064–5624, Doklady Mathematics, 2007, Vol. 75, No. 2, pp. 325–332. © Pleiades Publishing, Ltd., 2007.
Original Russian Text © Yu.V. Gulyaev, V.F. Kravchenko, V. I. Pustovoit, 2007, published in Doklady Akademii Nauk, 2007, Vol. 413, No. 3, pp. 320–328.

In this paper, we suggest and substantiate construc-
tions of a new class of WA-systems of Kravchenko–
Rvachev functions [1, 2] based on atomic functions
(AFs) and using the ideas and results of [1–7]. In the
first part, we describe an algorithm for constructing
WA-systems of functions, and in the second part, we
perform a computational study of a new family of com-
plex wavelets based on AFs and possessing a number of
useful properties. These wavelets and their Fourier
images are determined by analytical relations.
CONSTRUCTION OF KRAVCHENKO–RVACHEV
WAVELET FUNCTIONS
According to [1–3], the construction of a wavelet
function
ψ
is determined by the requirement that a large
number of the coefficients
zero. This number depends mainly on the regularity of
the function
f
, the number of zero moments of
the size of the support of
f
. If
ψ
has sufficiently many zero moments, then the wavelet
coefficients
〈
f
,
ψ
j
,
n
〉
are small if the scale
this case,
ψ
has
p
zero moments



〈

f

,

ψ

j

,

n

〉

must be close to



ψ

, and



f

is a regular function and












2

j

is small. In





(1)

Theorem 1.
a scaling function generating orthogonal bases. If
and

ϕ
(
ω
)

are

p
-
times continuously differentiable at the
frequency

ω
= 0,
then the following assertions are
valid
:
the wavelet

ψ

has

p

zero moments



Suppose that

ψ



and



ϕ



are a wavelet and



ψ

(

ω

)



























;
tkψ t ( ) t d
∞
–
+∞
∫
0,where 0
kp.
<≤
=

ψ
h

(
ω

ω

)



and its first

and its first


p

p

– 1
– 1

derivatives vanish at
derivatives vanish at
+∞
∑


ω

ω

= 0;
=


(


)










π

.
For any

0

≤



k

<

p q

k

(

t

) =

ϕ

(

t

–

n

)

is a poly-
nomial of degree
The size of the support.
larity at
t
0
and
1
2j

k

.


If the function

f

has a singu-
ψ
j
,




t

0

is inside the support of




n

(

t

) =

ψ

, then the amplitude of

〈

f

,

ψ

j

,

n

〉



may be
large. If
level
tain the point
cients, it is necessary to decrease the support of
following compact support theorem is valid [1, 3].
Theorem 2.
A scaling function
ported if and only if so is h
are equivalent. If the supports of

ψ

has compact support of size
contains
K
wavelets
t
0
. To reduce the number of large coeffi-

K

, then each

2

j




ψ

j

,

n

, whose supports con-




ψ

. The





ϕ



is compactly sup-



and the supports of

h

and

ϕ



and



h







ϕ

are equal to
N1
N2
–
2

[

N

1

,

N

2

],

then the support of



ψ equals
,
, where h is the dual mirror filter.
It follows from the properties of AFs [2] that their
Fourier spectra do not vanish at ω = π. Therefore, AFs
cannot be used as h(ω) (otherwise, one of the assertions
of Theorem 1 is false). To satisfy the conditions of The-
orem 1, we write h(ω) in the form h(ω) =
⎛
2--- -
⎝ ⎠
cos · (ω). Here, (ω) is the Fourier-spec-
trum of the AF. On the time domain, such a function
(e.g., the AF up(t)) can be represented by using the
Laplace transform. Some basic properties of the AF
up(t), which refer to problems of operational calculus,
are given in [1, 2]. In this case, h(ω) has p zero
moments for ω = π. Such an approach makes it possi-
ble to use AFs in constructing new classes of wavelet
functions. Since the scaling function ϕ(x) must be
orthonormal, h(ω) must have some special properties,
nk
n
∞
–=
--------
t
-------------- -
2jn
2j
–
⎝⎠
⎛⎞
1+
--------------------------- -
N2
--------------------------- -
N1
2
–1+
2 -
⎝
ω
⎛ ⎞⎠⎞
p
θ˜
θ˜
A New Class of WA-Systems
of Kravchenko–Rvachev Functions
Academician Yu. V. Gulyaeva, V. F. Kravchenkoa,
and Academician V. I. Pustovoitb
Received November 21, 2006
DOI: 10.1134/S1064562407020391
a Institute of Radio Engineering and Electronics, Russian
Academy of Sciences, ul. Mokhovaya 11, korp. 7, Moscow,
125009 Russia
e-mail: olegk@lianet.ru
b Central Design Bureau of Unique Instrumentation,
Russian Academy of Sciences, ul. Butlerova 15,
Moscow, 117342 Russia
COMPUTER
SCIENCE

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326
DOKLADY MATHEMATICS Vol. 75 No. 2 2007
GULYAEV et al.
namely, |h(ω)|2 + |h(ω + π)|2 = 1. According to [1, 3], if
a 2π-periodic function h(ω) has these special properties
1
2π()1/2
j
1=
and the product
(2–jω) converges, then its
limit ϕ(ω) belongs to L2(R), and ||ϕ
≤ 1. In this case,
the scaling function is defined by ϕ(ω) =
h(2–iω).
Having determined the scaling function ϕ(x), we can
find the wavelet generating function ψ. For example, ψ
can be defined as
or, equivalently,
Then, ψ(x) and ϕ(x) are compactly supported functions
from L2(R) satisfying the equations
where the hn are given in the form
The algorithm for constructing WA-systems of
Kravchenko–Rvachev functions is as follows.
(i) Transforming the AF spectrum by the formula
⎛
2--- -
⎝ ⎠
h(ω) = cos(ω).
(ii) Constructing a scaling function of the form
∞
∏
ϕ(ω) =
h(2–iω) with finitely many multipliers.
The number of multipliers must be larger than 2p,
where p is the number of nonzero moments.
(iii) Constructing a wavelet generating function on
the basis of the AF in the form
--------------- -
h
∞
∏
||L2
1
2
------ -
i
1=
∞
∏
ψ ω( )
1
2
------ -e
iω
2--- -
–
hω
⎝
2--- -
π
+
⎠
⎛ ⎞ϕω
2--- -
⎝ ⎠
⎛ ⎞
=
ψ x ( )
21–
()n
1–hn
–1–
( )ϕ tn
–
().
n
∞
–=
+∞
∑
=
ϕ x ( )
2
hnϕ 2xn
–
(),
n
∞
–=
+∞
∑
=
ψ x ( )
21–
()nhn
–1+ϕ 2xn
–
(),
n
∞
–=
∞
∑
=
h ω( )
1
2
------ -
hne
inω
–
.
n
∞
–=
+∞
∑
=
2 -
⎝
ω
⎛ ⎞⎠⎞
p
θ˜
1
2
------ -
i
1=
ψ ω ( )
1
2
------ -e
iω
2--- -
–
hω
⎝
2--- -
π
+
⎠
⎛ ⎞ϕω
2--- -
⎝ ⎠
⎛ ⎞,=
or, equivalently, ψ(x) = –1)n – 1h(–n – 1)ϕ(t – n),
where (ω) is conjugate to h(ω). The resulting wavelet
generating function must satisfy the following condi-
tions.
A. The zero mean condition:
(x)dx = 0. A
numerical experiment shows that it does hold.
B. The stability condition: A ≤ (2–jω)|2 ≤ B.
This condition also holds, because the function ψ(x) is
obtained by transforming the AF spectrum, which is
bounded from above by max(
decreases to zero. We do not multiply the AF by any
quantity removing the upper bound for the spectrum
amplitude. A numerical experiment confirms that this
condition is satisfied.
(ω)) = 1 and rapidly
C. The orthogonality condition: for ψj, k(x) =
2j/2ψ(2jx – k) and ψl, m(x) = 2l/2ψ(2lx – m), where j, k, l,
m ∈ Z, 〈ψj, k, ψl, m〉 = δj, lδk, m. A numerical experiment
shows that the Kravchenko–Rvachev wavelet functions
constructed above are orthogonal.
CONSTRUCTION OF COMPLEX
KRAVCHENKO–RVACHEV WAVELETS
Construction
Kravchenko–Rvachev wavelets [1] are based on the fol-
procedure. The complex
lowing convolution of an AF and the multiplier cosp
,
where p ∈ ?:
(2)
where (ω) is the spectrum of the AF. The spectrum of
the family of complex wavelets (without normaliza-
tion) has the form
(3)
To find an analytical representation for the wavelets, we
must determine the inverse Fourier transform for
expression (3). For this purpose, consider the inverse
ω
2--- -
Fourier transform of cosp
. We use the following rep-
resentation for odd p:
2(
n
∞
–=
+∞
∑
h
ψ
∞
–
∞
∫
|
j
∞
–=
∞
∑
ψ)
up
ω
2--- -
hˆθ
pω( )
ω
2--- -
cos
⎝⎠
⎛⎞
p
θˆω( ),=
θˆ
ψ ˜ˆθ
p
hˆθ
pω(π
–
).=

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DOKLADY MATHEMATICS Vol. 75 No. 2 2007
A NEW CLASS OF WA-SYSTEMS OF KRAVCHENKO–RVACHEV FUNCTIONS 327
(4)
where
=
. For even p, we have
(5)
It is known that
(6)
According to (4)–(6), for odd p, we have
(7)
and for even p, we have
(8)
Let us find the inverse Fourier transform for (2) by
using (7) and (8). For even p, we obtain
(9)
Similarly, for odd p, we have
(10)
It follows from (3) that
ω
2--- -
cosp
1
2p
1–
---------- -
pω
2--- -
⎝⎠
⎛⎞
cos
Cp
1
p
2–
()ω
2--- -
⎝⎠
⎛⎞-
cos+=
+ Cp
2
p
4–
()ω
2--- -
⎝⎠
⎛⎞
cos
…
Cp
p
----------- -
1–
2
ω
2--- -
⎝ ⎠
⎛ ⎞
cos++,
Cp
k
p!
k! p
(
k
–
)!
----------------------- -
ω
2--- -
⎝ ⎠
⎛ ⎞
cosp
1
2p
1–
---------- -
pω
2--- -
⎝⎠
⎛⎞
cos
Cp
1
p
2–
( )ω
2--- -
⎝⎠
⎛⎞----- -
cos+=
+ Cp
2
p
4–
( )ω
2--- -
⎝⎠
⎛⎞
cos
…
Cp
p
----------- -
2–
2
ω
2--- -
⎝ ⎠
⎛ ⎞
cos++
1
2p
---- -Cp
p
2-- -
.+
F
1–
aω()
cos
{}
δ t
(
------------------------------------------- -.
a
+
)δ t
(
a
–
)
+
2
=
F
1–
ω
2--- -
⎝ ⎠
⎛ ⎞
cosp
⎩⎭
⎨⎬
⎧⎫
= 1
2p
---- -
Cp
kδ t
p
-------------- -
2k
–
2
+
⎝⎠
⎛⎞
δ t
⎝
p
-------------- -
2k
–
2
–
⎠
⎛⎞
+,
k
0=
p
2–
∑
()/2
F
1–
ω
2--- -
⎝ ⎠
⎛ ⎞
cosp
⎩⎭
⎨⎬
⎧⎫
1
2p
---- -
Cp
kδ t
p
-------------- -
2k
–
2
+
⎝⎠
⎛⎞
k
0=
p
2–
∑
()/2
=
+ δ t
p
-------------- -
2k
–
2
–
⎝⎠
⎛⎞
Cp
p
2-- -
δ t ( )
+.
hθ
pt ( )
1
2p
---- -
Cp
kθ t
⎝
p
-------------- -
2k
–
2
+
⎝⎠
⎛⎞⎛
k
0=
p
2–
∑
()/2
=
+ θ t
p
-------------- -
2k
–
2
–
⎝⎠
⎛⎞⎠⎞
Cp
p
2-- -
θ t ( )
+.
h˜θ
pt ( )
= 1
2p
---- -
Cp
kθ t
⎝
p
-------------- -
2k
–
2
+
⎝⎠
⎛⎞
θ t
⎝
p
-------------- -
2k
–
2
–
⎠
⎛⎞
+
⎠
⎛⎞.
k
0=
p
1–
∑
()/2
(11)
(12)
Therefore, the complex Kravchenko–Rvachev wavelets
are determined up to a normalizing multiplier by
expressions (9)–(12). However, in practice, wavelets
must have unit norm in the space L2. Let us determine
the norm of function (11). For this purpose, we calcu-
late ||
(t)|| with taking into account the fact that the
shift of a function does not affect its norm in the space
L2. After transformations, we obtain the following
expression for even p:
(13)
Similarly, for odd p, we have
(14)
⎞
A shift of a function in the frequency domain does not
affect its L2-norm either; therefore,
eiπthθ
=
(15)
The expression for the above family of Kravchenko–
Rvachev wavelets with unit norm has the form
1
hθ
(16)
where
The Fourier transform of wavelets (16) is
(t) is determined by (12).
(17)
where
(ω) is determined by (2).
ψ ˜θ
pt ( )
eiπthθ
pt ( ),=
hθ
pt ( )
hθ
h˜θ
pt ( ),
p is even,
pt ( ),
p is odd.
⎩
⎨
⎧
=
hθ
p
hθ
pt ( )
1
2p
---- - 2
Cp
k
()
2θ t ( )
2
Cp
p/2
()
2θ t ( )
2
+
k
0=
p
2–
∑
()/2
⎝
⎜
⎛
=
+ 2
Cp
kCp
p/2θ t
p
-------------- -
2k
–
2
+
⎝⎠
⎛⎞θ t ( )
⎝⎠
⎛⎞
k
0=
p
2–
∑
()/2
∞
(
–
∞
∫
+
Cp
kCp
p/2θ t
p
-------------- -
2k
–
2
–
⎝⎠
⎛ ⎞θ t ( )
⎝⎠
⎛⎞
k
0=
p
2–
∑
)/2
+
Cp
iCp
jθ t
p
------------- -
2i
–
2
+
⎝⎠
⎛ ⎞θ t
p
-------------- -
2j
–
2
–
⎝⎠
⎛⎞
j
0=
p
2–
∑
()/2
i
0=
p
2–
∑
()/2
2
t d
⎠
⎟
⎞
1/2
.
h˜θ
pt ( )
1
2p
---- - 2
Cp
k
()
2θ t ( )
2
k
0=
p
1–
∑
()/2
⎝
⎜
⎛
=
+ 2
Cp
iCp
jθ t
p
------------- -
2i
–
2
+
⎝⎠
⎛⎞θ t
p
-------------- -
2j
–
2
–
⎝⎠
⎛⎞
j
0=
p
1–
∑
()/2
i
0=
p
1–
∑
()/2
2
t d
⎠
⎟
1/2
.
ψ ˜θ
pt ( )
pt ( )
hθ
pt ( ) .=
ψθ
pt ( )
pt ( )
----------------eiπthθ
pt ( ),=
hθ
p
ψ ˆθ
pω( )
2π
pω( )
hˆθ
------------------ -hˆθ
pω(π
–
),
p
?,
∈
=
hˆθ
p

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DOKLADY MATHEMATICS Vol. 75 No. 2 2007
GULYAEV et al.
Graphs of normalized Kravchenko–Rvachev com-
plex wavelets and their Fourier transforms are shown in
Figs. 1–3.
Substantiation. Let us show that functions (16)
obtained above are wavelets. For this purpose, we must
prove that they satisfy the conditions [4]
(18)
(19)
If (18) holds, then condition (19) is equivalent to
(20)
(21)
It follows from (9)–(12) that the wavelets under con-
sideration are products of linear combinations of
finitely many shifted AFs and the bounded continuous
function eiπt. Therefore, they inherit the following prop-
erties of the AFs from which they are obtained: conti-
nuity, compact support, and boundedness. These prop-
erties ensure the fulfillment of conditions (18) and (20)
(that they have unit norm is ensured by the above nor-
malization of function (11)). Thus, it remains to show
that functions (16) satisfy the zero mean condition (21).
This follows from the vanishing of the Fourier trans-
form (17) at zero:
(22)
ψθ
p
L2,
ψθ
p
∈
1;=
2π
ψ ˆθ
------------------- a
pa
( )
a
2
d
∞
–
∞
∫
Cψ
∞.
<
=
t ψθ
pt ( ) t d
∞
–
∞
∫
∞;
<
ψθ
pt ( ) t d
∞
–
∞
∫
0, or, equivalenty=
ψ ˆθ
p0
( )
0.=
ψ ˆθ
p0
( )
1
pt ( )
π
2-- -
⎠
hθ
----------------hˆ
π
–
()
=
=
1
pt ( )
hθ
----------------
cos
⎝
⎛⎞
p
θˆ
π
–
()
0.=
PROPERTIES OF COMPLEX
KRAVCHENKO–RVACHEV WAVELETS
In this section, we consider properties of the new
family of Kravchenko–Rvachev wavelets.
Zero mean. As shown above (see (21), (22)), the
complex Kravchenko–Rvachev wavelets satisfy the
zero mean condition
Smoothness. The complex Kravchenko–Rvachev
wavelets have the same degree of smoothness as the
AFs from which they are obtained. Indeed, wavelets
(16) are (up to multiplication by a constant) products of
eiπt, which is an infinitely differentiable function, and
the function
(t), which is the sum of weighted shifts
by the argument of the AF θ(t). Therefore, for infinitely
differentiable AFs, complex wavelets (16) are infinitely
differentiable as well.
The size of the support. The complex Kravchenko–
Rvachev wavelets are compactly supported, because so
are AFs. The support of wavelets (16) coincides with
that of the function
(t), which, in turn, depends on
the support of the AFs. The form of the dependence is
easy to determine by using (9)–(11). Suppose that the
AF θ(t) has support suppθ(t) = [a, b] and the support of
its complex wavelet is
(23)
It follows from (23) that the size of the support of a
wavelet increases with its order p. The sizes of the sup-
ports of some complex Kravchenko–Rvachev wavelets
are given in Tables 1–3.
Frequency resolution. The frequency resolution of a
wavelet ψ(t) is defined as the reciprocal of its frequency
spread [4]:
(24)
ψθ
pt ( ) t d
∞
–
+∞
∫
0.=
hθ
p
hθ
p
suppψθ
pt ( )
a
p
2-- -
–
b
p
2-- -
+
,
.=
σω
2
1
2π
----- -
ωη
–
()2ψ ˆ ω( )
2ω,
d
0
∞
∫
=
Table 1. The frequency spreads and supports of the complex Kravchenko–Rvachev wavelets (t) based on the AFs upm(t)
p
up1(t) up2(t) up3(t) up4(t)up5(t)
suppsuppsupp suppsupp
1
2
3
4
5
1.78
1.15
0.85
0.67
0.56
[–1.5, 1.5]
[–2, 2]
[–2.5, 2.5]
[–3, 3]
[–3.5, 3.5]
1.56
0.97
0.71
0.56
0.46
[–1.5, 1.5]
[–2, 2]
[–2.5, 2.5]
[–3, 3]
[–3.5, 3.5]
1.33
0.83
0.60
0.47
0.39
[–1.5, 1.5]
[–2, 2]
[–2.5, 2.5]
[–3, 3]
[–3.5, 3.5]
1.19
0.74
0.54
0.43
0.35
[–1.5, 1.5]
[–2, 2]
[–2.5, 2.5]
[–3, 3]
[–3.5, 3.5]
1.17
0.74
0.54
0.42
0.35
[–1.5, 1.5]
[–2, 2]
[–2.5, 2.5]
[–3, 3]
[–3.5, 3.5]
ψθ
p
σω
2
σω
2
σω
2
σω
2
σω
2

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DOKLADY MATHEMATICS Vol. 75 No. 2 2007
A NEW CLASS OF WA-SYSTEMS OF KRAVCHENKO–RVACHEV FUNCTIONS329
0.8
0.6
0.4
0.2
0
–0.2
–0.4
–0.6
–0.8
–1.5 –1.0
1.0
0 0.51.0 1.5
ψ(t)
up1
–0.5
0.8
0.6
0.4
0.2
0
–0.2
–0.4
–0.6
–0.8
0.8
0.6
0.4
0.2
0
–0.2
–0.4
–0.6
–0.8
0.8
0.6
0.4
0.2
0
–0.2
–0.4
–0.6
–0.8
0.8
0.6
0.4
0.2
0
–0.2
–0.4
–0.6
–0.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
–0.2
–0.4
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
–0.2
1.8
2.0
1.5
1.0
0.5
0
–0.5
2.0
1.5
1.0
0.5
0
–0.5
2.0
1.5
1.0
0.5
0
–0.5
2.5
–1012–2
–1012 –2
–1012–2–33
–1012–2–33
t
–15 –10051015–5
–15 –1005 1015–5
–15 –1005 10 15 –5
–15 –1005 1015–5
–15 –1005 1015
ω
–5
up2
up3
up4
up5
ψ(ω)
Fig. 1. Left: the graphs of complex Kravchenko–Rvachev wavelets ψ(t) based on the AFs up(t) of orders 1–5 (the dashed lines show
their imaginary parts); right: the corresponding Fourier transforms
(ω) of the wavelets.
ψ ˆ