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Iterated Function Systems, Moments, and Transformations of Infinite Matrices
About this Title
Palle E. T. Jorgensen, Department of Mathematics, The University of Iowa, Iowa City, IA 52242-1419, U.S.A., Keri A. Kornelson, Department of Mathematics, University of Oklahoma, Norman, OK 73019-0315, U.S.A. and Karen L. Shuman, Department of Mathematics and Statistics, Grinnell College, Grinnell, IA 50112-1690, U.S.A.
Publication: Memoirs of the American Mathematical Society
Publication Year:
2011; Volume 213, Number 1003
ISBNs: 978-0-8218-5248-4 (print); 978-1-4704-0620-2 (online)
DOI: https://doi.org/10.1090/S0065-9266-2011-00627-1
Published electronically: March 2, 2011
Keywords: Moments,
measure,
iterated function system,
moment matrix,
Hankel matrix distribution,
fractals,
orthogonal polynomials,
operators in Hilbert space,
Hilbert matrix,
positive definite functions,
spectral representation,
spectral measures,
rank-one perturbations,
spectrum,
absolutely continuous,
Hermitian operator,
self-adjoint operator,
unbounded operator,
Hilbert space,
deficiency indices,
self-adjoint extension
MSC: Primary 28A12, 34B45, 42C05, 42A82, 46E22, 47L30, 47L60, 47C10, 47S50, 54E70, 60J10, 60J20, 78M05, 81S30, 81Q15, 81T75, 82B44, 90B15
Table of Contents
Chapters
- Preface
- 1. Notation
- 2. The moment problem
- 3. A transformation of moment matrices: the affine case
- 4. Moment matrix transformation: measurable maps
- 5. The Kato-Friedrichs operator
- 6. The integral operator of a moment matrix
- 7. Boundedness and spectral properties
- 8. The moment problem revisited
- Acknowledgements
Abstract
We study the moments of equilibrium measures for iterated function systems (IFSs) and draw connections to operator theory. Our 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, we 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, our 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.
We first examine the classical existence problem for moments, culminating in a new proof of the existence of a Borel measure on $\mathbb {R}$ or $\mathbb {C}$ with a specified list of moments. Next, we consider moment problems associated with affine and non-affine IFSs. Our main goal is to determine conditions under which an intertwining relation is satisfied by the moment matrix of an equilibrium measure of an IFS. Finally, using the famous Hilbert matrix as our prototypical example, we study boundedness and spectral properties of moment matrices viewed as Kato-Friedrichs operators on weighted $\ell ^2$ spaces.
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