The classical moment problem and some related questions in analysis

Akhiezer N.I.

This book explains the classical moments problem, which can be applied to reconstruct distribution functions, electrical conductivities tensors…in plasmas, partially ionized gases, etc.

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Table of contents :
Series……Page 1
Title page……Page 2
Date-line……Page 3
FOREWORD……Page 4
CONTENTS……Page 8
1. Basic Concepts……Page 10
2. Properties of the Polynomials associated with a Jacobi Matrix……Page 17
3. Theorems of Invariance and Analyticity……Page 24
4. Quadrature Formula. Continued Fractions……Page 29
Addenda and Problems……Page 33
1. Solubility Criteria……Page 38
2. The Isometric Operator generated by Orthogonal Polynomials……Page 43
3. Some Criteria of Completeness……Page 51
4. The Function $rho(z)$ and the Nevanlinna Matrices……Page 58
5. Extremal Properties of the Functions $rho_n(z)$ and $rho(z)$……Page 69
6. M. Riesz’ Method……Page 77
Addenda and Problems……Page 88
1. An Interpolation Problem in the Theory of Analytic Functions……Page 99
2. Reduction of the Power Moment Problem to a certain Problem in the Theory of Functions……Page 104
3. An Algorithm for Consecutive Linear Fractional Transformations……Page 110
4. Canonical Solution of the Indeterminate Hamburger Problem……Page 122
Addenda and Problems……Page 133
1. The Operator Approach to the Moment Problem……Page 147
2. Symmetric Operators which can be represented with the Aid of $mathcal{J}$-Matrices……Page 155
3. Integral Representations of a Positive Functional……Page 162
Addenda and Problems……Page 170
1. The Trigonometric Moment Problem……Page 187
2. Orthogonal Polynomials on a Circle……Page 191
3. Hermitian-Positive Functions of a single Argument……Page 199
4. Hermitian-Positive Functions in many-dimensional Spaces……Page 203
5. Absolutely Monotonic and Exponentially Convex Functions……Page 212
Addenda and Problems……Page 221
Appendix. Stieltjes Continued Fractions……Page 241
Bibliography……Page 252
Index……Page 260