Revista
de la
Universidad
del Zulia
Fundada en 1947
por el Dr. Jesús Enrique Lossada
DEPÓSITO LEGAL ZU2020000153
ISSN 0041-8811
E-ISSN 2665-0428
Ciencias
Exactas,
Naturales
y de la Salud
Año 14 N° 40
Mayo - Agosto 2023
Tercera Época
Maracaibo-Venezuela
REVISTA DE LA UNIVERSIDAD DEL ZULIA. 3ª época. Año 14, N° 40, 2023
Bulatnikova Irina Vyacheslavovna // Method of Mathematical Theory of Moments, 84-102
DOI: https://doi.org/10.46925//rdluz.40.05
84
Method of Mathematical Theory of Moments
Bulatnikova Irina Vyacheslavovna*
ABSTRACT
Infinite matrices play an important role in many aspects of analysis, algebra, differential
equations, and the theory of mechanical vibrations. Jacobi matrices are interesting because
they are the simplest representatives of symmetric operators in infinite-dimensional space.
they are used in interpolation theory, quantum physics, moment problem. In this paper,
based on the elements of Jacobi matrix, it will be determined the type of the operator that
occurs when processing the results of measurements of random variables. The first type of
operators are matrices, for which the moment problem has a unique solution, and Jacobi
matrix generates a specific moment problem. The second type of operators are matrices, for
which the moment problem has many solutions, and Jacobi matrix is said to generate an
indeterminate moment problem.
KEY WORDS: Infinite matrices, Jacobi matrices, moment problem, type of operator.
*Moscow Aviation Institute (National Research University) MAI. ORCID: https://orcid.org/0000-
0002-5514-8198. E-mail abcdeik@mail.ru
Recibido: 18/01/2023 Aceptado: 09/03/2023
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Bulatnikova Irina Vyacheslavovna // Method of Mathematical Theory of Moments, 84-102
DOI: https://doi.org/10.46925//rdluz.40.05
85
Método de la Teoría Matemática de los Momentos
RESUMEN
Las matrices infinitas juegan un papel importante en muchos aspectos del análisis, el
álgebra, las ecuaciones diferenciales y la teoría de las vibraciones mecánicas. Las matrices de
Jacobi son interesantes porque son las representantes más simples de los operadores
simétricos en el espacio de dimensión infinita; se utilizan en teoría de interpolación, física
cuántica, problema de momento. En este trabajo, con base en los elementos de la matriz de
Jacobi, determinaremos el tipo de operador que se presenta al procesar los resultados de las
mediciones de variables aleatorias. El primer tipo de operadores son las matrices, para las
cuales el problema de momento tiene una solución única, y la matriz de Jacobi genera un
problema de momento específico. El segundo tipo de operadores son las matrices, para las
cuales el problema de momento tiene muchas soluciones, y se dice que la matriz de Jacobi
genera un problema de momento indeterminado.
PALABRAS CLAVE: Matrices infinitas, Matrices de Jacobi, problema de momentos, tipo de
operador.
Introduction
Infinite matrices play an important role in many aspects of analysis, algebra,
differential equations, and theory of mechanical vibrations. They are also connected with
one type of algebraic continued fractions widely applied by classics of XIX century, and,
first of all, by P.L. Chebyshev. In particular, the so-called systems of orthogonal polynomials
were introduced through the mediation of these fractions (Akhiezer, 1961).
Infinite matrices can be found in different branches of mathematics (Dyukarev,
2015). They were initially studied in the theory of summation of divergent sequences and
series, in quantum mechanics and theory of solving infinite systems of linear equations with
finite number of unknown variables. For example, the transformations specified with the
help of infinite matrix А = (aij) are considered in the theory of series.
The solutions of two linear equations in infinite matrices are used in Heisenberg-
Dirac theory in quantum mechanics: AX−XA=1 and AX−XD=0 (the first equation is called
the quantization equation). The operator spectrum theory in Hilbert space is used to find
the solutions.
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Algebraic properties of infinite matrices and infinite-dimensional linear or classical
groups are investigated in many papers and monographs. This is done from many points of
view, among which the theory of associative rings and modules, algebraic K-theory, theory
of Lie algebras and algebraic groups, theory of infinite groups, functional analysis (operator
rings, spectral analysis), elemental analysis (theory of functions, sequences and series),
theory of representations, theory of models, infinite combinatorial analysis and probability
theory can be highlighted.
Infinite matrices are summed up as ordinary matrices. But when multiplying infinite
matrices, their specific character is revealed. Namely, the multiplication of infinite matrices
is not always specified. In analysis, in which complex-valued and real-valued infinite
matrices are used, this situation is overcome by applying conditions of convergence of
coefficients in lines and columns to the matrices. Matrices with the coefficients from
arbitrary ring R with one are considered in algebra, thus, other finitude conditions, finite-
lineness and finite-columnness type are applied. Besides, multiplication can be specified
but, at the same time, be non-associative. Third, the invertibility of infinite matrices has its
specific character – for example, there are infinite matrices with infinite number of
reciprocals.
Jacobi matrices can occur in different mathematical problems (continued fraction
theory, differential equations). Mathematical models of elemental processes, the physical
nature of which is known, are written down as formulas and dependencies known for these
processes. As a rule, static problems are expressed as algebraic expressions, dynamic – as
differential or finite-difference equations. At the same time, any differential equation has an
infinite number of solutions in partial derivatives.
The solution method often consists in transition to non-stationary problems, which
are approximated by the systems of finite-difference equations. In practice, the solutions
satisfying the additional conditions are of the most interest. As a rule, the problems
describing physical or chemical processes in the frameworks of differential equations in
partial derivatives comprise the boundary conditions.
Jacobi matrices are of interest since they are the simplest representatives of
symmetrical operators in infinite-dimensional space . They are used in interpolation
theory, quantum physics, moment problem.
The term “moment problem” was for the first time found in the paper of 1894-1895 by
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T. Stieltjes. However, one important problem, related to moment problem, had already been
set and, in particular case, solved by P.L. Chebyshev as early as in 1873. P.L. Chebyshev
referred to his problem several times during the whole last period of his life. However, in his
problem P.L. Chebyshev saw, first of all, the way to obtain some limiting theorem of
probability theory. The comprehensive study of the problem and its different
generalizations are the merit of A.A. Makarov.
Based on Jacobi matrix elements it is possible to recognize the operator type
occurring while processing the measuring results of random variables, i.e. it is possible to
understand the classification of operators generated by Jacobi matrices in two-dimensional
space. In particular, it is necessary to be able to recognize the operator type by elements of
Jacobi matrices. The types, to be found further, can be of two types.
The first type comprises such matrices, for which the corresponding moment
problem has a unique solution, therefore, it is said that Jacobi matrix, in this case, generates
a certain moment problem.
The second type comprises such cases when the moment problem has many
solutions. In this case, the moment problem is called indeterminate.
The problem is urgent and sparks interest with specialists in different branches of
mathematics. Finding the operator class with the help of infinite Jacobi matrices can serve
as a new solution for some aspects.
1. Definitions and terms
Let С be a set of complex numbers. Let us designate Hilbert space of infinite sequences
of vectors u=( through , where ∊С and ν=( ,, …) , where ∊С, i= 0, 1,2, … .
As it is known, u∊, if series and scalar product of elements u, ν∊
are found by the equality: (u, ν) = . Thus, is completely separable Hilbert space.
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Vector u∊ is called finite, if it has a finite number of non-zero components.
We will call the finite matrix
Jacobi matrix, in which ак are real and bк are positive, k=0 , 1 ,2, … .
Matrix J by the operations
j=0, 1 ,2, … , (1)
where
defines on the variety of all finite vectors, i.e. vectors with a finite number of non-zero
coordinates of space, the nonclosed symmetric operator whose closure we designate
through L.
The linear operator L is called symmetric in Hilbert space Н, if its definition region
DL is dense in Н and if the following equality is relevant for any
u
,
v
(finite vectors):
(L
u
,
v
)=(
u
, L
v
)
It is known that L is a minimum closed symmetric operator generated by the
expression (1) and boundary condition = 0 in space , generally speaking, it is not self-
conjugated and defect indices п+ and п- of this operator equal the pairs (0, 0) or (1, 1), i.e. п+ =
п- =1 or п+ =п- =0.
Let A be a symmetric operator. Let us consider the homogeneous equation Ах=λх,
where is linear space of solutions of this equation. It can be proved that if А is a
symmetric operator and λ is an arbitrary complex number from the upper half-plane
(Imλ>0), then dimensionality does not change, i.e. it is the same for any λ. This number
(dimensionality NA) is called the upper defective number. The lower defective number is, in
this case, when λ is taken from the lower half-plane (Im λ <0).
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The main theorems, their application for classifying operators and facts given below are
taken from the book (Akhiezer, 1961), (Suetin, 1979), (Refaat El Attar, 2006), (Koralov,
Sinai, 2013), papers (Kostyuchenko, Mirzoev, 1998), (Kostyuchenko, Mirzoev, 2001),
(Korepanov, 1999) and other literature sources.
2. Main theorems and their application for classifying operators
The main theorems and facts given below are taken from the book (Akhiezer, 1961;
Koralov, 2013; Suetin, 1979), papers (Koralov, 2013; Kostyuchenko, Mirzoev, 1998;
Korepanov, 1999) and other literature sources.
The finite-difference equation
has two linearly independent solutions: (λ)and (λ), corresponding to the initial
conditions (λ)=1 , (λ)= , (λ)=0, (λ)=
Solution (λ) of this equation we seek in the form: ( λ) = скχк(λ), k = 0,1,...,
where cк does not depend on λ and is defined for the conditions to be fulfilled
b
к
c
к
+1 = -b
к
- 1 c
к
- 1
, k=0,1,... (2)
and с0 = , с1 = 1. From these conditions we obtain by the mathematical induction
method:
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So, Ck can be written down as follows:
(3)
Let us substitute uj for Pj (λ) in the formula (1)
, if
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Taking into account the equality (2), we have:
where
Let us demonstrate that the following formula is correct
First, we will prove the formula (5) for k = 2j
Then
Let us consider the case for k = 2j – 1
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Then
We assured ourselves that the formula (5) is correct. Taking into account the formula
(4), we have:
Now let us consider the equality:
We select
Let us apply the mathematical induction method, we will take j as odd
Let us sum up all the obtained equalities. We have:
Let us express
We do the same for all even j. We have:
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The following theorems are true.
Theorem 1.1 (see (Koralov, 2013), theorem 2.1)
Let elements of matrix j are such that
where sequence ck is defined by the following equality:
Then there is quite determinate case for matrix j.
Proof:
Let M be the space of limited sequences of matrices with dimensionality p x p with norm
where and i is a natural number.
Let us define the expression of by the following equalities:
and
b
, if
, if
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and element u(0) by the following equalities:
Thus, u(0)
∊
M. Besides, from the definition of operator F1 and identities:
it follows that sequence is a fixed point of the expression of
On the other hand, number i can be selected so for the expression of Fi to be contractive.
Actually, if, for example, then
These inequalities follow from the properties of absolute value, definition of the
expression of Fi and equality
Let us fix 0<p<1. From the demonstrated inequalities and theorem conditions it follows
that number i can be taken so large that with all the following inequality is
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fulfilled:
which was to be proved. So, with fixed sequence belongs
to space M. Further, from this fact, equalities
and condition a) of the theorem it is seen that
with any
The conclusion is derived from this theorem.
Conclusion 1.1 (see (Koralov, 2013), conclusion 2.1)
Let the elements of matrix J to be such that
Then there is quite indeterminate case for matrix J.
Proof.
First, let us demonstrate that where j = 1, 2, … (6)
Actually, from the formula
, if
, if
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and condition 1) it follows that, for example,
Applying the inequalities (6) we find that condition 1) of theorem 1.1 is fulfilled, since
condition b) of theorem 1.1 is fulfilled based on condition 2), since
based on condition 3).
Consequently, if the elements of Jacobi matrix satisfy conditions 1) – 3), the assertion of
theorem 1.1 is true.
Let us take some nonreal point λ and construct the sequence of circles for it Kn (λ). Since
there is either a limit circle or a limit point K∞ (λ).
Theorem 1.3 (see (Akhiezer, 1961), theorem 1.3.1)
a) With any nonreal λ there is at least one solution of the equation
for which
i.e. the solution, which belongs to ℓ2.
b) Any solution of this equation belongs to ℓ2, only if K∞ (λ) is a circle.
Theorem 1.2 (see (Koralov, 2013), theorem 1.3.2)
If K∞ (λ) is a circle for some nonreal λ, then K∞ (λ) will be a circle for any nonreal λ.
Moreover, if the series
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converges in some nonreal point of λ, then it converges uniformly in each finite part of
complex λ-plane.
J matrix is called a matrix of C type, if there is a case of limit circle for it, and of D type,
if there is a case of limit point for it. Defect indices for matrices of D type will be n+ = n- = 0,
and for matrices of C type – n+ = n- = 1.
3. Main results of using the method of classifying operators by elements of
infinite Jacobi matrix
Example 1.
Jacobi matrix with elements an = a∙nα, bn = b∙nβ is given. What relations between a, b, α, β
must be for theorem 1.1 or conclusion 1.1 to be fulfilled?
Solution.
Let us consider for conclusion 1.1 assuming that n=k.
1. Let us check the fulfillment of the first condition:
This is true for any positive b, β.
2. Let us check the fulfillment of the second condition, i.e. the series
converges.
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This can be proved with the help of integral Cauchy criterion on the series
convergence.
This series converges at β > 1.
3.
So, Jacobi matrix with elements an = a∙nα and bn = b∙nβ, where a, b – any positive numbers,
α < β – 1 and β > 1 fulfills condition 1.1, i.e. there is an indeterminate case and defect indices
will be (1, 1).
Carleman theorem
If series then there is a determinate case for such matrix, i.e. defect indices
will be (0, 0)
(regardless of αk).
If in our example β ≤ 1, then it follows from Cauchy integral criterion that series
diverges. Thus, applying Carleman theorem, we have that in the band 0 < β ≤ 1 and α any of
our Jacobi matrices has the defect index (0, 0).
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Example 2.
Jacobi matrix with elements ak = a∙kk and bk = bk is given, where a, b, α – given numbers.
What relations between a, b, α must be for theorem 1.1 or conclusion 1.1 to be fulfilled?
Solution.
Let us check if the conditions of conclusion 1.1 are fulfilled.
1.
This is fulfilled for any b.
2. Let us consider series
Using Cauchy criterion on series convergence, we found out that this series converges
at b > 1.
3. Let us consider series Let us apply D'Alembert criterion once again
We found out that series converges at any a, α and b > 1.
We came to the conclusion that Jacobi matrix with elements ak = a∙kk and bk = b, where a,
α – any numbers, b > 1, fulfills conclusion 1.1, i.e. there is an indeterminate case and defect
indices will be (1, 1).
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Example 3.
Jacobi matrix with elements ak = ak and bk = b∙kβ is given. What relations between a, b, β
must be for theorem 1.1 and conclusion 1.1 to be fulfilled?
Solution.
Let us check if the conditions of the problem for conclusion 1.1 are fulfilled.
1. Let us write down the first condition of the conclusion: Let us substitute
bk = b∙kβ. We have
This is fulfilled for any b and β.
2. Let us apply Cauchy integral criterion to prove the second condition of the
conclusion
From this we have that – β + 1 < 0, i.e. β > 1.
3. Let us check the fulfillment of the third condition of conclusion 1.1, substituting
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instead of ak = ak and bk = b∙kβ. We obtain series . Let us apply D'Alembert criterion
on the series convergence
For the given series to converge, it is necessary that a < 1.
We came to the conclusion that Jacobi matrix with elements ak = ak and bk = b∙kβ fulfills
conclusion 1.1 under the condition that a < 1, β > 1 and b – any number. Then there is an
indeterminate case, the defect indices will be the same (1, 1).
Conclusion
Finding the operator class with the help of infinite Jacobi matrices can serve as a new
solution for some aspects, which are rather urgent and which spark interest with
specialists in different branches of mathematics.
As a result, based on Jacobi matrixes it is possible to recognize the operator type
generated in space by these matrices. It was found out that they can be of two types.
The first type is D (limit point case) – the moment problem has a unique solution
and it is called determinate. The second type is С (limit circle case) – it occurs when the
moment problem has many solutions. In this case, the moment problem is called
indeterminate. It was also revealed that the defect indices for C type are п
_=
п
+=
1 and for D
type – п
_=
п
+ =
0. In the abovementioned examples it is demonstrated how it is possible to
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use theorem 1.1, Carleman theorem and conclusion 1.1 to define the operator classification.
Mathematical models, in which systems of linear equations are used, are
widely applied in different fields, including the processing of results of random
variables. For example, in Markovian processes used in queue theory.
The moment method in mathematical statistics is one of the most general
methods of finding statistical estimates for unknown parameters of distributing
probabilities based on the observation results.
References
Akhiezer, N.I. (1961). Classical moment problem and some analysis aspects connected with
it. — State edition of physical and mathematical literature, Moscow.
Dyukarev, Yu. M. (2015). Block Jacobi matrices and matrix problem of Hamburger
moments. — Scientific news, series: Mathematics. Physics, №5(202), issue 3.
Koralov, L. B.; Ya.G., (2013). Sinai. Probability theory and random processes (translated
from English by E.V. Perekhodtseva. — Moscow: Moscow Center of Continuous
Mathematical Education, 407 p.
Korepanov, I. G. (1999). Fundamental mathematical structures of integrated models. —
Theory of mathematical physics, 01/1999; 118(3):405-412. DOI:10.4213/tmf713.
Kostyuchenko, A. G.; Mirzoev, K. A. (1998). Three-term recurrences with matrix
coefficients. Quite indeterminate case. — Mathematical Notes, volume 63, issue 5, pp. 709-716.
Kostyuchenko, A. G.; Mirzoev, K. A. (2001). Features of quite indeterminateness of Jacobi
matrices with matrix elements. — Functional analysis and its applications, volume 35, issue
4, pp. 32-37.
Refaat El Attar, (2006). Special Functions and Orthogonal Polynomials. — Lulu Press,
Morrisville NC 27560.
Suetin, P. K. (1979). Classical orthogonal polynomials. — Nauka, 2nd edition, 1979.
