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2023 | Buch

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Jacobi Matrices and the Moment Problem

verfasst von: Yurij M. Berezansky, Mykola E. Dudkin

Verlag: Springer Nature Switzerland

Buchreihe : Operator Theory: Advances and Applications

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This monograph presents the solution of the classical moment problem, the construction of Jacobi matrices and corresponding polynomials. The cases of strongly,trigonometric, complex and real two-dimensional moment problems are discussed, and the Jacobi-type matrices corresponding to the trigonometric moment problem are shown. The Berezansky theory of the expansion in generalized eigenvectors for corresponding set of commuting operators plays the key role in the proof of results.
The book is recommended for researchers in fields of functional analysis, operator theory, mathematical physics, and engineers who deal with problems of coupled pendulums.

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Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

For a better understanding of the content of the monograph, we will first outline the main content of the second chapter: direct and inverse spectral problems for classical Jacobi matrices and orthogonal polynomials on the real axis.

Yurij M. Berezansky, Mykola E. Dudkin

Chapter 2. Some Aspects of the Spectral Theory of Unbounded Operators

Abstract

We hope that the reader is well acquainted with basic courses of mathematical and functional analysis, in particular with the theory of linear bounded operators in Hilbert spaces. Therefore, in this chapter we will recall only the basic concepts regarding unbounded self-adjoint operators, without which the book reading would be difficult.

Yurij M. Berezansky, Mykola E. Dudkin

Chapter 3. Jacobi Matrices and the Classical Moment Problem

Abstract

This section is devoted to the presentation of the basic provisions of the theory of (ordinary numerical tri-diagonal) Jacobi matrices and the classical moment problem which is closely related to this theory.

Yurij M. Berezansky, Mykola E. Dudkin

Chapter 4. The Strong Moment Problem

Abstract

In this chapter, we present the main provisions concerning the strong moment problem. More precisely, strong classical, when studying the possibility of representing a sequence of real numbers \((s_n)\) in the integral form for all integers moments.

Yurij M. Berezansky, Mykola E. Dudkin

Chapter 5. Block Jacobi Type Matrices in the Complex Moment Problem

Abstract

This chapter considers the generalization of the connection between the classical moment problem and the spectral theory of self-adjoint Jacobi matrices. Namely, an analogue of the Jacobi matrix is proposed, which is related to the complex moment problem and also to the system of polynomials orthogonal with respect to some measure with a compact support on the complex plane. Such a matrix has a block tri-diagonal structure and is a normal operator acting in the spaces of square summable sequences \(l_2\). The existence of a one-to-one correspondence between such measures on the complex plane and block tri-diagonal Jacobi-type normal matrices is proved. For the sake of simplicity, only the bounded normal operator is considered. From the point of view of the complex moment problem, this restriction means that the measure associated with orthogonal polynomials has a compact support. The chapter ends with some solution of the complex moment problem.

Yurij M. Berezansky, Mykola E. Dudkin

Chapter 6. Unitary Block Jacobi Type Matrices and the Trigonometric Moment Problem

Abstract

This chapter considers yet another, in comparing with Chap. 5, generalization of the connection between the classical moment problem and the spectral theory of self-adjoint Jacobi matrices. Namely, an analogue of the Jacobi matrix is proposed for consideration, which related to the trigonometric moment problem, as well as the system of polynomials orthogonal with respect to some probability measure with a support on the unit circle. Such a matrix also has a block tri-diagonal structure and generates a unitary operator that acts in the \(l_2\) space like the space of sequences summable with a square. By using this connection, the existence of a one-to-one correspondence between probability measures on the unit circle and the block tri-diagonal Jacobi type unitary matrices is established.

Yurij M. Berezansky, Mykola E. Dudkin

Chapter 7. Block Jacobi Type Matrices and the Complex Moment Problem in the Exponential Form

Abstract

This chapter considers the generalization of the information given in Chaps. 3 and 5. Namely, an analogue of the Jacobi type block matrix related to the complex moment problem in the exponential form is proposed and corresponding polynomials, orthogonal with respect to some probability measure on the complex plane are investigated. In this case, two commutative block matrices are obtained, which both have a tri-diagonal block structure, one of them generates a unitary and the second one generates a self-adjoint operator in the space like the space of square summable sequences \(l_2\)-type space. A one-to-one correspondence is also established between probability measures on a compact set of the complex plane and such couple of matrices.

Yurij M. Berezansky, Mykola E. Dudkin

Chapter 8. Block Jacobi Type Matrices and the Two Dimensional Real Moment Problem

Abstract

The chapter proposes an analogue of Jacobi type bock matrices, which are related to the real power two-dimensional moment problem and the corresponding polynomials orthogonal with respect to some probability measure on the real plane. In this case, two commutative block matrices are also obtained, which have a tri-diagonal block structure and are self-adjoint operators in the \(l_2\) space like the space of square summable sequences. The existence of a one-to-one correspondence between the probability measure on the compact set of the real plane and such matrices is also proved.

Yurij M. Berezansky, Mykola E. Dudkin

Chapter 9. Applications of the Spectral Theory of Jacobi Matrices and Their Generalizations to the Integration of Nonlinear Equations

Abstract

In this chapter, the difference analogue of the well-known procedure for finding solutions of the Cauchy problem for some nonlinear partial differential equations, in particular the Korteweg-de-Vries (KdV) equation, is presented, using the inverse spectral problem for the Sturm-Liouville equation on the half-axis.

Yurij M. Berezansky, Mykola E. Dudkin

Backmatter

Titel: Jacobi Matrices and the Moment Problem
verfasst von: Yurij M. Berezansky
Mykola E. Dudkin
Copyright-Jahr: 2023
Verlag: Springer Nature Switzerland
Electronic ISBN: 978-3-031-46387-7
Print ISBN: 978-3-031-46386-0
DOI: https://doi.org/10.1007/978-3-031-46387-7

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