Continuum Wavelets and Distributions
YURI K. DEM'YANOVICH, OLGA N. IVANTSOVA, ALEKSANDRA Y. PONOMAREVA
Parallel Algorithm Department
Saint Petersburg State University
University nab.,7/9, St. Petersburg
RUSSIA
Abstract: - The purpose of this work is to obtain a wavelet expansion of information flows, which are
distribution flows (in the terminology of Schwartz). The concept of completeness is introduced for a family of
abstract functions. Using the mentioned families, nested spaces of distribution flows are constructed. The
projection of the enclosing space onto the nested space generates a wavelet expansion. Decomposition and
reconstruction formulas for the above expansion are derived. These formulas can be used for wavelet
expansion of the original information flow coming from the analog device. This approach is preferable to
the approach in which the analog flow is converted into a discrete numerical flow using quantization and
digitization. The fact is that quantization and digitization lead to significant loss of information and distortion.
This paper also considers the wavelet expansion of a discrete flow of distributions using the Haar type
functions.
Key-Words: - wavelets, information flows, distributions, calibration relations
Received: September 24, 2021. Revised: May 21, 2022. Accepted: June 23, 2022. Published: July 14, 2022.
1 Introduction
The processing of numerical information flows with
classical and non-classical wavelets have been
studied in a large number of works. Research
wavelet decompositions for flows of a more
complex nature (flows of matrices, p-adic numbers,
etc.) were mainly based on the theory of non-
classical wavelets.
Wavelet decomposition is one of the main
means of the processing of numerical information
flows. Let us give several examples of the
application of these expansions in technology and
medicine. In research [1] the separate models for
signal de-noising with different ratio signal/noise
were designed. The discrete wavelet
decompositions were used. The result was applied to
the computerized analysis of Lung Sound.
Paper [2] is devoted to the damage severity
quantification of the brain by using a wavelet
packet. The proposed technique shows significant
benefit in compressing spatio-spectral patterns of
multichannel signals in just a unified visual
representation.
The timely and high-quality maintenance of
electrical networks is a prerequisite for their trouble-
free operation.
In work [3], complex wavelets are used for creating
an efficient algorithm for such processing. The
proposed algorithm achieves higher accuracy with
reduced training time in the classification of events
than compared to the reported event classification
methods. To date, there are several studies on the
theory of wavelets, among which deserve special
mention works by I. Daubechies [4], C. Chui [5],
S. Mallat [6], Yu. N. Subbotin and Chernykh [7],
I. Ya. Novikov, V. Yu. Protasov and M. A. Skopina
[8]. Research in this field also includes a series
of modern works. Cubic wavelets with two zero
moments are obtained in work [9]. Five-diagonal
splitting for cubic splines with six zero moments on
the segment was obtained in [10]. Paper [11] deals
with structural issues concerning wavelet frames
and their dual frames. In paper [12] the authors
define the wavelet multiplier and Landau-Pollak-
Slepian operators on the Hilbert space. In paper
[13] the wavelet optimized finite difference B-spline
polynomial chaos method is proposed. The method
is applied to the solution of stochastic partial
differential equations. In paper [14] the authors
propose a highly efficient and accurate valuation
method for exotic-style options based on the novel
Shannon Wavelet Inverse Fourier Technique
(SWIFT).
These studies mainly reflect the classic
approach to wavelets, which is based on various
variants of the Fourier transform, applied to the
multiple-scale ratio to obtain a scaling function
and ultimately wavelet decomposition. However,
the practice of processing numeric flows required
expanding the framework of the classical theory.
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W.Sveldens constructed a lifting scheme for an
area that is not invariant relative to the shift. The
concept of non-stationary wavelets, introduced by
I.Ya. Novikov, also led to the expansion
framework of the mentioned theory. The need to
significantly speed up computations was faced
with great theoretical difficulties that arose on the
path of the development of the classical approach
to the wavelet expansions (see [15] – [27]). In
paper [18] the authors propose an algorithm with a
high level of confidentiality while maintaining high
image quality. Paper [19] presents a powerful, fast
and reliable signal analysis method based on the
massively parallel continuous wavelet transform
algorithm. The nonlinear wavelet estimates of the
spectral densities for non-Gaussian linear processes
are considered in paper [20]. The paper [21]
presents an efficient algorithm based on the
Galerkin method using biorthogonal Hermite
multiwavelets with cubic splines. The authors of
paper [22] propose an effective approach to
obtaining approximate solutions of linear and
nonlinear two-dimensional Volterra integro-
differential equations. with usage of two-
dimensional wavelets. In [23], to solve the problems
of low contrast and fuzzy boundary in the
traditional wavelet transform, a threshold function
is proposed. Paper [24] presents a new structure for
a single-pixel image using compression probing in
shift-invariant spaces by using the sparsity property
of the wavelet representation. In [25] case studies of
typical nonlinear de-noising problems in various
domains are conducted. Study [26] focused on the
classification of Electroencephalography signal. The
study aims to make a classification with fast
response and high-performance rate. Paper [27]
proposes and
discusses a new Electroencephalography de-
noising technique, based on a combination of
wavelet transforms and conventional filters.
The listed works show the wide use of wavelets
in various fields of human activity. They apply to
physics, chemistry, biology and medicine. In most
cases these are the results of a large number of
measurements at some points in space and at certain
points in time. In fact the mentioned measurements
are neither a point nor instantaneous. This fact, long
noticed, led to the theory of Schwartz distributions.
Along with value streams ordinary functions should
also be considered distribution flows. In this regard,
the use of distributions is more natural, since such
an approach reflects the idea of a trial function.
Mentioned flows of distributions can be continuous
or discrete. In this and in another case, their wavelet
decomposition is important, allowing the more
efficient use of computer and communication
resources.
The purpose of this work is to study information
flows associated with certain trajectories in
distribution spaces. Elements of the spaces are the
mentioned trajectories, whose parameters take the
values from a set of non-zero Lebesgue measure.
For these trajectories (also called the families of
distributions) the concept of completeness is
introduced. The complete family is used to build
a space of distributions. The criterion of
embedding of the mentioned spaces is discussed.
The projection of the enclosing space on an
embedded space generates a wavelet decomposition.
It is shown that from the considered continuum
case, we can pass to a discrete case. As a result of
the transition, we obtain spaces of the Haar-type
functions. In this case, the mentioned embedding
criterion becomes calibration ratios.
2 Generating Function
Let , be measurable sets of non-zero Lebesgue
measure on real axis. Let , where is an
open set of the real axis. Consider
linear space of basic functions (in this
case we assume that is the standard linear
space of main functions). Thus the space
consists of all infinitely differentiable and
compactly supported functions , , i.e. such
that .
The space of distributions (the space dual to
) denote . The relevant duality is
denoted by sharp brackets, namely, the result
action of the distribution on the main function
v is denoted by ,v.
Let be a family of distributions from the
space , where is a family parameter, . A
family of this kind is called a
trajectory in
(or an abstract function with values
in ).The expression is called the trajectory
component. For the record of trajectory components
it is sometimes convenient to use square brackets,
setting . For the main function , the
expreson is an ordinary
function of the argument defined on measurable
set .
Let , . Consider the set all
trajectories with property
We denote this set by .
Lemma 1. The following statements are true.
1. The set (1) is not empty.
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2. There are trajectories ,
which do not lie in the set .
3. The set is a linear space.
Proof. 1. Consider the trajectory
, where is the delta-function
at the point . In this case
is continuous function in , so
. So, it is established that the set
is not empty.
2. Let be a point of an open interval,
contained in . Consider the trajectory
, where ,.
Then(x).
Remaining in set , choose a main function such
that () and choose so that . In this
case, the function does not belong to space
. The second part of the lemma is proved.
3. If and are two elements of the set
, then by definition the functions
and lie in the space . We have
It follows that . The third part of
the lemma is proved.
It is obvious that
Therefore, for , we
can discuss the integral
Let us introduce the notation
, (3)
Since for and function
lies in the space , so
from (3) -- (4) for expressions (2) we have
. (5)
In what follows, we will sometimes use a
shorter notation without mention of the main
function For example, conditions (5) can be
written in the form
. (6)
Here and below, the presence of the main function
is implied (see (5) -- (6)).
In what follows, we will need the notion of a
complete abstract function.
In this connection, we first introduce the concept
of a complete family of mappings.
Definition 1. Let be a linear topological
space, be the dual space, be a non-empty set
of parameters . For every fixed we consider a
mapping :
H
. The family of the mappings
is called complete in H if for the
condition follows the equality
, i.e.
In particular, if is a set of numbers, then
is called an abstract function with values in
(or
a trajectory in ). In this case, if family
is complete, then it is called the complete abstract
function in
(or the complete trajectory in ).
Let be some set on the real axis,
Consider the complete trajectory
in the space . In this way
and
1) for every fixed the function of
argument is an element of space ,
,
2) the relation
An example of a complete trajectory in
for the case when =(0,1), ={0,1,2,…} is the
family .
Consider the linear space Ʋ defined by the
relation
Note that according to the accepted notation, the
formula contains the main function
implicitly, so that the mentioned formula is
equivalent to formula. Thus the
space consists of distributions.
Lemma 2. For any element of the space
there is a unique family distributions ,
such that .
Proof. We will prove by contradiction.
Suppose there is an element , , which has two
representations
where and are two families from the space
. By (9) the identity follows
Introducing the notation
from (10) we get relation (7), so that . So the
families and are the same.
The resulting contradiction proves the assertion.
This concludes the proof.
3 Embedded Space
Here we consider an analog of previous construction
with replacement of the set by the set ,
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In particular,
=
,
(
(11)
Let ℜ be a linear operation from to
For a distribution taking into
account the previous agreement for
we have .
Using (11), for we have
ℜw
ℜw
(12)
Formulas of the form (12) will sometimes be
written in the form
ℜw
. (13)
Suppose ,
is a complete family in . Consider the linear
space
defined by the relation
Note that according to the accepted notation, the
formula
contains the main function
implicitly, so that the mentioned formula is
equivalent to formula
, .
Thus the space
consists of distributions.
Consider the linear operation Ƥ, which acts from
space into the space
,
Let's suppose that
Theorem 1. If the relations (15) -- (16) are right,
then the space (14) is contained in the space (8),
Proof. According to formula (14), for the element
a fair representation is
Using representation (16) in (18), we have
In view of the obvious relationship
we get
where belongs to the
space .
From the definition (8) of the space it is clear
that the distribution (19) is an element of this
space, . Formula (17) has been established.
This completes the proof.
4 Wavelet Decomposition
Let condition (16) be satisfied. According to
Theorem 1 the space
is embedded in the space ,
i.e. relation (17) holds.
Consider the projection operation of the space
onto the space
,
(20)
According to Lemma 2, for the element
there are unique elements and
such that
the next representations hold
Thus the element
is uniquely defined by the
element . Appropriate map is denoted
by ,
It is easy to see that is the linear operation
acting from the space into the space
,
.
From (21) -- (22) it follows, that operation
is defined by the operation according to the
formula
Theorem 2. For any element ,
( the ratio
(24)
is right.
Proof. Since formula (23) is equivalent to formula
then, taking into account relation (16), from (25) we
find
Using the notation adopted in (13) -- (14), we see
that relation (26) leads to equality (24).
This completes the proof.
Let us introduce the operation ,
where is the identity operation in . As a result
of projection (20) we obtain the direct sum
where
, .
Consider . Let's put
.
Theorem 3. For relations
are fulfilled. Here .
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Proof. From (24) -- (27) we have
Thus, relation (28) is valid. From formulas (22)
and (27) we obtain relation (29).
This concludes the proof.
The element is the initial flow, the element
is the main flow and the element is the wavelet
flow. Formulas (22), (27) are called
decomposition formulas, and formulas (29) is called
reconstruction formulas.
We introduce a linear operation
by formula
(30)
Theorem 4. For the operation defined by formula
(23) to be the projection operation of the space
onto the space
it is necessary and sufficient to
have
.
(31)
Here is the identical operation in the
space
Proof. Necessity. Let be a projection operation
onto the space
. Then the idempotency condition
is satisfied: . In other words, on elements
of the space
the operation acts as the identical
operation,
On the other hand, by the definition of the
operation we have
(33)
Using the definition of the operation
(see
formula (30)) from (33) we obtain
(34)
Setting , by (34) we find
(35)
The obvious transformation of the left side of
relation (35) gives us the formula
(36)
Using property (16) on the left side of formula
(36), we get the equality of the left sides of
relations (32) and (36). Therefore identity
(37)
is right. In view of the completeness of the family
we derive relation (31) by formula (37).
The necessity has been proven.
Sufficiency. Assume that relation (31) holds. The
definition of the operation given by formula (23),
shows that for an element we have
.
Notice, that in view of the notation (30), formula
(23) is equivalent to formula (34).
Let be an arbitrary element of the space
. In
(30) we take As a result,
we get
(38)
In view of assumption (31), from relation (38) we
easily find
(39)
It follows from (39) that the operation is
idempotent. The sufficiency of relation (31) has
been established.
This concludes the proof.
5 Integral Operation Case
Here we give an illustration of the previous
situations where is an integral operation.
Let be a function of two arguments
such that the integral operation with
kernel ,
, (40)
maps the space to the space ,
Consider two abstract functions
{ {
which are complete in the spaces and
, respectively.
Let's suppose that
Theorem 5. If relation (42) holds, then
. (43)
Proof. According to formula (14), for the element
the representation is true
<=><
(44)
Relation (44) is equivalent to the formula
<
(45)
In view of formula (40), condition (42) can be
rewritten as
Using representation (46) in relation (45), we have
<
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Rearranging the order of integration in (47) leads
to the formula
<
According to condition (41), the expression in
square brackets is an element of the space
Thus, in accordance with formula (8) relation (48)
is a representation for element of the space
This completes the proof.
6 Space of the Haar Type
Let be an interval . Consider a grid
Let's put where
is defined by grid (49) -- (50),
If is fixed in the interval , then there is
so that . When so fixed the expression
is piecewise constant function of the
argument . This the function is equal to the
constant ( for and
equals zero for .
Thus, for every fixed it is obvious that
implication is correct.
Let be the space of piecewise constant
functions, which are constants on each interval
, For functions , from the space
we introduce the notation
Consider dual spaces
and
.
Relevant duality can be defined by the formula
It is easy to check that is a complete
trajectory in . We introduce the notation
for . From (51) we
get
By definition we put ,
For we have
By (53) we reduce
=
=
where
The convergence of the series and integrals
appearing here is obvious.
Consider the linear space of trajectories in the
distribution space by setting
In view of formulas (54) -- (56) we have
Denote by the linear space of locally
summable
functions,
w(t)=
Let us assume that the condition
( (59)
is right. For a finite interval condition (59)
is always satisfied.
Theorem 6. Under condition (59) family
{ linear homomorphisms of the space
into the space S exists.
Proof. By Holder's inequality, we have
|
Hence we have
In view of the condition we can see
the sum
is finite. So we have
Consider the mapping of the space into the
space by matching the element ,
an element
, (62)
where for a fixed the expressions
are numbers. From relations (57), (59) and
(61) - (62) follows the implication . The
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linearity of the mapping is obvious. This
completes the proof.
7 Embedding of the Haar Type
Spaces
Next, we assume that
. (63)
Consider the function given by the
formula
(64)
where
(65)
We define the linear operator by the relations
(66)
By (63) -- (66) we get
We substitute the function (see
formula (51)) in (67). We put ,
For we have
If , then in the first integral values
and are in the same the same grid interval, namely,
in the interval According to formula
(51) in this integral the integrand is equal to
(. The second integral (68) under
these conditions is equal to zero. Thus,
Similarly, we find
It is easy to see that
From (69) -- (71) it follows that for function
value on the interval is unit, and
outside this interval its value is zero.
For swe have
(72)
If in (72) the variable is in the interval,
then the integral of functions over the
mentioned interval is left in the last sum. In view
of formula (51), the result of integration turns out
to be equal to one. Thus, throughout the entire
interval function is equal to
unit, and outside this interval it is equal to zero,
(73)
Introducing the notation for
, from (51) and (73) we obtain the
calibration relations
Consider the linear space
trajectories in the
distribution space by setting
By definition
From (73) and (76) we have
=
(77)
where
The convergence of the series and integrals
appearing in (77) is obvious.
Thus, the space (75) can be represented in the
form
Let be the linear space of locally summable
functions,
|
where , .
Let us assume that the condition
<+
are right.
Theorem 7. Under condition (80) the following
statements are true:
1. Linear spaces
and are subspaces of the
spaces Ʋ and S respectively.
2. Under condition (80) there exists a family
{
linear homomorphisms of the space
into the space .
Proof. The first assertion follows from
calibration relation (74). The proof of the second
assertion is carried out similarly to the proof of
Theorem 6.
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In the space , consider a new coordinate system
, obtained from system (52) by multiplying the
coordinate functionsinto constants .
We have
,
Calibration relation (74) takes the form
, (82)
where
In the space , consider the functionals and
given by the formulas
,
The properties of biorthogonality are easily
verified
,
where is the Kronecker symbol.
Projection of from onto define by
functionals (84) -- (85),
Let
.
Operation (86) derives a wavelet decomposition of
space ,
.
(87)
Consider an element in basis of the
space ,
(88)
.
(89)
Let's be known the coefficients and of
projections for the element onto spaces and
in (87),
u
u
(90)
where
,
Let's express the numbers through the numbers
and . According to the formulas (82) and (90)
we have the representation
On the other hand, representation (88) -- (89) is
valid. Equating the right parts of representations
(88) and (91) taking into account the linear
independence of the coordinate splines
leads to ratios
. (92)
Relations (92) are formulas of reconstruction.
We introduce vectors
as well as the matrix Ƥ=(, whose elements
are given by formula (83).
The introduced notation allows us to write the
formulas reconstruction (92) in the form
The vector
is the initial flow,
is main
flow, and the vector
is wavelet flow.
Consider Using
equalities
we have
From (92) we successively derive
Using the notation
we get
Formulas (95) -- (96) are called formulas
decomposition.
Introducing the matrix we
rewrite the decomposition formulas in the form
,
Note that the constants are calculated from
formulas (65) and (83), while the numbers are
determined by formulas (85), (94).
Referring to the projection of the space on
,
note that their structure is similar the structure of
the spaces and , respectively.
According to formulas (57) and (58) we have
w(t)=
In the same way, the structures of the spaces
(78) and (79) are similar,
,
|
To construct a wavelet expansion in the case of
spaces and
we just use obtained formulas for
the mentioned decomposition of the spaces and
.
For this purpose, we introduce the vectors
8 Projection onto Embedded Space
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= (…,
= (…,
c = (…,
whose components are elements of the space .
We assume that
, and
Theorem 8. The wavelet expansion for spaces
consists of decomposition formulas
, b
and reconstruction formulas
c
The element c is called the initial distribution flow,
element
is called basic distribution flow, and is
called wavelet flow of distributions.
9 Discussion
In this paper, we first consider the space of basic
functions (i.e., infinitely differentiable compactly
supported functions) and the space of
distributions.
Then we discuss the set of trajectories
in the whose action on any basic function lies
in the space , . It is proved that
is not empty, and it is a linear space. For
example, a trajectory of the distributions ,
belongs to .
In addition, we consider abstract functions of
real variable with values in the space ,
The notion of a complete abstract function in
is introduced. Let be a complete
abstract function in . An example of a
complete function is for =(1,2),
. We consider a space of
distribution trajectories which are generated by the
function . The space , embedded in the
space is constructed similarly. A projection
operation of the space onto generates
wavelet decomposition of the space . From here
decomposition and reconstruction formulas for
distribution flows are obtained. These formulas can
be used for wavelet decompositions of the
information flows coming from analog devices.
This approach is preferable to situations where the
analog flow turns into a discrete numerical flow
with the help of quantization and digitization. The
point is that quantization and digitization lead to
significant loss of information and to distortion.
Therefore, it is preferable to carry out the wavelet
decomposition of the original analog flow.
However in certain cases it is required to process
discrete flows of distributions. This is not difficult
to achieve using special generating function
options. One of these options is also presented in
this work. In the case under consideration, we
arrive at the Haar-type coordinate functions. As a
result we obtain the decomposition and
reconstruction formulas corresponding to this case
for discrete flow of distributions.
An important issue is the adaptive choice of
embedded space and projection operation for it. In
the case of ordinary functions, this problem is
solved by the appropriate consolidation of the initial
divisions (see [15] – [17]). Such a choice can be
made by the use of local approximation properties
for functions in one or another metric space. For
distribution flows, similar issues are to be
considered in the future.
10 Conclusion
In this paper, a new approach to the construction of
wavelet expansions is considered. This approach
allows us to consider discrete and continuous
flows of distributions. Such flows arise in many
physical problems. As it is known, point actions
considered in theory, in fact actually do not exist.
In this regard, the use of distributions is more
natural, since such an approach reflects the idea of
a trial function. Mentioned flows of distributions
can be continuous or discrete. In that and in
another case, their wavelet decomposition is
important, allowing the more efficient use of
computer and communication resources. This
work shows the possibility of wavelet
decomposition of both continuous and discrete flow
of distributions. In the future, it is planned to use
the distributions to study the spaces of dipoles and
the conditions for their embedding. We suppose to
obtain a wavelet decomposition of the dipole spaces.
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WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2022.21.62
Yuri K. Demyanovich, Olga N. Ivantsova,
Aleksandra Y. Ponomareva
E-ISSN: 2224-2880
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Volume 21, 2022
