Skip to main content
SpringerLink
Log in

Probability and Fourier Duality for Affine Iterated Function Systems

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

Let d be a positive integer, and let μ be a finite measure on ℝd. In this paper we ask when it is possible to find a subset Λ in ℝd such that the corresponding complex exponential functions e λ indexed by Λ are orthogonal and total in L 2(μ). If this happens, we say that (μ,Λ) is a spectral pair. This is a Fourier duality, and the x-variable for the L 2(μ)-functions is one side in the duality, while the points in Λ is the other. Stated this way, the framework is too wide, and we shall restrict attention to measures μ which come with an intrinsic scaling symmetry built in and specified by a finite and prescribed system of contractive affine mappings in ℝd; an affine iterated function system (IFS). This setting allows us to generate candidates for spectral pairs in such a way that the sets on both sides of the Fourier duality are generated by suitably chosen affine IFSs. For a given affine setup, we spell out the appropriate duality conditions that the two dual IFS-systems must have. Our condition is stated in terms of certain complex Hadamard matrices. Our main results give two ways of building higher dimensional spectral pairs from combinatorial algebra and spectral theory applied to lower dimensional systems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Conze, J.-P., Hervé, L., Raugi, A.: Pavages auto-affines, opérateurs de transfert et critères de réseau dans R d. Bol. Soc. Bras. Mat. (N.S.) 28(1), 1–42 (1997)

    Article  MATH  Google Scholar 

  2. Dutkay, D.E., Han, D., Picioroaga, G., Sun, Q.: Orthonormal dilations of Parseval wavelets. Math. Ann. 341(3), 483–515 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. Dutkay, D.E., Jorgensen, P.E.T.: Wavelets on fractals. Rev. Mat. Iberoam. 22(1), 131–180 (2006)

    MATH  MathSciNet  Google Scholar 

  4. Dutkay, D.E., Jorgensen, P.E.T.: Iterated function systems, Ruelle operators, and invariant projective measures. Math. Comput. 75(256), 1931–1970 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Dutkay, D.E., Jorgensen, P.: Oversampling generates super-wavelets. Proc. Am. Math. Soc. 135(7), 2219–2227 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Dutkay, D.E., Jorgensen, P.E.T.: Analysis of orthogonality and of orbits in affine iterated function systems. Math. Z. 256(4), 801–823 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  7. Dutkay, D.E., Jorgensen, P.E.T.: Fourier frequencies in affine iterated function systems. J. Funct. Anal. 247(1), 110–137 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dutkay, D.E., Jorgensen, P.E.T.: Martingales, endomorphisms, and covariant systems of operators in Hilbert space. J. Oper. Theory 58(2), 269–310 (2007)

    MATH  MathSciNet  Google Scholar 

  9. Dutkay, D.E., Røysland, K.: The algebra of harmonic functions for a matrix-valued transfer operator. J. Funct. Anal. 252(2), 734–762 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30(5), 713–747 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  11. Jorgensen, P.E.T., Kornelson, K.A., Shuman, K.L.: Affine systems: asymptotics at infinity for fractal measures. Acta Appl. Math. 98(3), 181–222 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Jorgensen, P.E.T.: Analysis and Probability: Wavelets, Signals, Fractals. Graduate Texts in Mathematics, vol. 234. Springer, New York (2006)

    MATH  Google Scholar 

  13. Jorgensen, P.E.T., Pedersen, S.: Spectral theory for Borel sets in R n of finite measure. J. Funct. Anal. 107(1), 72–104 (1992)

    Article  MathSciNet  Google Scholar 

  14. Jorgensen, P.E.T., Pedersen, S.: Group-theoretic and geometric properties of multivariable Fourier series. Expo. Math. 11(4), 309–329 (1993)

    MathSciNet  Google Scholar 

  15. Jorgensen, P.E.T., Pedersen, S.: Harmonic analysis and fractal limit-measures induced by representations of a certain C *-algebra. J. Funct. Anal. 125(1), 90–110 (1994)

    Article  MathSciNet  Google Scholar 

  16. Jorgensen, P.E.T., Pedersen, S.: Estimates on the spectrum of fractals arising from affine iterations. In: Fractal Geometry and Stochastics, Finsterbergen, 1994. Progr. Probab., vol. 37, pp. 191–219. Birkhäuser, Basel (1995)

    Google Scholar 

  17. Jorgensen, P.E.T., Pedersen, S.: Dense analytic subspaces in fractal L 2-spaces. J. Anal. Math. 75, 185–228 (1998)

    Article  MathSciNet  Google Scholar 

  18. Jorgensen, P.E.T., Pedersen, S.: Dense analytic subspaces in fractal L 2-spaces. J. Anal. Math. 75, 185–228 (1998)

    Article  MathSciNet  Google Scholar 

  19. Jorgensen, P.E.T., Pedersen, S.: Orthogonal harmonic analysis of fractal measures. Electron. Res. Announc. Am. Math. Soc. 4, 35–42 (1998)

    Article  MathSciNet  Google Scholar 

  20. Kahane, J.-P.: Géza Freud and lacunary Fourier series. J. Approx. Theory 46(1), 51–57 (1986). Papers dedicated to the memory of Géza Freud

    Article  MATH  MathSciNet  Google Scholar 

  21. Kigami, J.: Local Nash inequality and inhomogeneity of heat kernels. Proc. Lond. Math. Soc. (3) 89(2), 525–544 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kigami, J., Strichartz, R.S., Walker, K.C.: Constructing a Laplacian on the diamond fractal. Exp. Math. 10(3), 437–448 (2001)

    MATH  MathSciNet  Google Scholar 

  23. Lapidus, M.L., Neuberger, J.W., Renka, R.J., Griffith, C.A.: Snowflake harmonics and computer graphics: numerical computation of spectra on fractal drums. Int. J. Bifurc. Chaos Appl. Sci. Eng. 6(7), 1185–1210 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  24. Okoudjou, K.A., Strichartz, R.S.: Weak uncertainty principles on fractals. J. Fourier Anal. Appl. 11(3), 315–331 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  25. Rudin, W.: Fourier Analysis on Groups. Interscience Tracts in Pure and Applied Mathematics, vol. 12. Interscience, New York (1962)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Central Florida, 4000 Central Florida Blvd., P.O. Box 161364, Orlando, FL, 32816-1364, USA

    Dorin Ervin Dutkay

  2. Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City, IA, 52242-1419, USA

    Palle E. T. Jorgensen

Authors
  1. Dorin Ervin Dutkay
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Palle E. T. Jorgensen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dorin Ervin Dutkay.

Additional information

Research supported in part by a grant from the National Science Foundation DMS-0704191.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dutkay, D.E., Jorgensen, P.E.T. Probability and Fourier Duality for Affine Iterated Function Systems. Acta Appl Math 107, 293–311 (2009). https://doi.org/10.1007/s10440-008-9384-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-008-9384-2

Keywords

Mathematics Subject Classification (2000)

Search

Navigation