Expansions for Integrals Relative to Invariant Measures Determined by Contractive Affine Maps

  • Charles A. Micchelli
Conference paper
Part of the Springer Series in Computational Mathematics book series (SSCM, volume 19)


We discuss expansions for integrals to invariant measures of certain stationary Markov chains determined by contractive affine maps. In the homogeneous case, Appell polynomials generated by the Fourier transform of the invariant measure determines the expansion. Some facts about the spectral radius of a stationary subdivision operator and the Lipshitz class of refinable functions are also included.


Invariant Measure Entire Function Spectral Radius Iterate Function System Strong Operator Topology 
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  1. [1]
    Barnsley, M.F. and Demko, S. Iterated Function Systems and the Global Construction of Fractals, Proc. of the Royal Society of London A, 399(1985), 245–275.MathSciNetGoogle Scholar
  2. [2]
    Barnsley, M.F., Demko, S., Elton, J., and Geronimo, J., Markov Processes Arising from Function Iteration with Place Dependent Probabilities, Annales de l’Institute Henri Poincaré, 24, No. 3, (1988), 367–394.MathSciNetzbMATHGoogle Scholar
  3. [3]
    Cavaretta, A.S., Dahmen, W., and Micchelli, C.A., Stationary Subdivision, IBM Research Report No. 15194, 1989, to appear in Memoirs of AMS.Google Scholar
  4. [4]
    Dahmen, W. and Micchelli, C.A., Stationary Subdivision and the Construction of Orthonormal Wavelets, in “Multivariate Approximation and Interpolation”, ISNM 94, N. Haussmann and K. Jetter (eds.), Birkhauser Verlag, Basel, (1991), 69–90.Google Scholar
  5. [5]
    Demko, Stephen, Euler, Maclauren Type Expansions for Some Fractal Measures, preprint.Google Scholar
  6. [6]
    Hutchinson, J. Fractals and Self-Similarity, Indiana J. Math, 30(1981), 713–747.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    Micchelli, C.A. and Prautzsch, H., Uniform Refinement of Curves, Linear Algebra and Applications, 114/115(1989), 841–870.MathSciNetCrossRefGoogle Scholar
  8. [8]
    Micchelli, C.A. and Prautzsch, H., Refinement and Subdivision for Spaces of Integer Translates of a Compactly Supported Function in Numerical Analysis, edited by Griffiths, D.F., and Watson, G.A., (1987), 192–222.Google Scholar
  9. [9]
    Yosida, K., Functional Analysis, Springer-Verlag, Berlin, 1966.Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1992

Authors and Affiliations

  • Charles A. Micchelli
    • 1
  1. 1.Mathematical Sciences DepartmentIBM T.J. Watson Research CenterYorktown HeightsUSA

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