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Iterated Function Systems, Moments, and Transformations of Infinite Matrices
Palle E. T. Jorgensen, University of Iowa, Iowa City, IA, Keri A. Kornelson, University of Oklahoma, Norman, OK, and Karen L. Shuman, Grinnell College, IA
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Memoirs of the American Mathematical Society
2011; 105 pp; softcover
Volume: 213
ISBN-10: 0-8218-5248-5
ISBN-13: 978-0-8218-5248-4
List Price: US$70
Individual Members: US$42
Institutional Members: US$56
Order Code: MEMO/213/1003
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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

  • Notation
  • The moment problem
  • A transformation of moment matrices: the affine case
  • Moment matrix transformation: measurable maps
  • The Kato-Friedrichs operator
  • The integral operator of a moment matrix
  • Boundedness and spectral properties
  • The moment problem revisited
  • Acknowledgements
  • Bibliography
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