# Iterated Function Systems, Moments, and Transformations of Infinite Matrices

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*Palle E. T. Jorgensen; Keri A. Kornelson; Karen L. Shuman*

The authors study the moments of equilibrium measures for iterated function systems (IFSs) and draw connections to operator theory. Their main object of study is the infinite matrix which encodes all the moment data of a Borel measure on \(\mathbb{R}^d\) or \(\mathbb{C}\). To encode the salient features of a given IFS into precise moment data, they establish an interdependence between IFS equilibrium measures, the encoding of the sequence of moments of these measures into operators, and a new correspondence between the IFS moments and this family of operators in Hilbert space. For a given IFS, the authors' aim is to establish a functorial correspondence in such a way that the geometric transformations of the IFS turn into transformations of moment matrices, or rather transformations of the operators that are associated with them.

#### Table of Contents

# Table of Contents

## Iterated Function Systems, Moments, and Transformations of Infinite Matrices

- Preface vii8 free
- Chapter 1. Notation 112 free
- Chapter 2. The moment problem 920
- Chapter 3. A transformation of moment matrices: the affine case 2132
- Chapter 4. Moment matrix transformation: measurable maps 3142
- Chapter 5. The Kato-Friedrichs operator 4152
- Chapter 6. The integral operator of a moment matrix 5364
- Chapter 7. Boundedness and spectral properties 6374
- Chapter 8. The moment problem revisited 8394
- 8.1. The shift operator and three incarnations of symmetry 8394
- 8.2. Self-adjoint extensions of a shift operator 8596
- 8.3. Self-adjoint extensions and the moment problem 8798
- 8.4. Jacobi representations of matrices 90101
- 8.5. The triple recursion relation and extensions to higher dimensions 96107
- 8.6. Concrete Jacobi matrices 98109

- Acknowledgements 101112
- Bibliography 103114