Analysis and Mathematical Physics

, Volume 1, Issue 1, pp 69–99 | Cite as

Geometry of spectral pairs

Article

Abstract

In this paper we develop a geometric framework for spectral pairs and for orthogonal families of complex exponentials in L2(μ), where μ is a given Borel probability measure supported in \({\mathbb {R}^{d}}\).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Billingsley, P.: Probability and measure. Wiley, New York. Wiley Series in Probability and Mathematical Statistics (1979)Google Scholar
  2. 2.
    Bohnstengel J., Kesseböhmer M.: Wavelets for iterated function systems. J. Funct. Anal. 259(3), 583–601 (2010)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Dutkay D.E., Han D., Jorgensen P.E.T.: Orthogonal exponentials, translations, and Bohr completions. J. Funct. Anal. 257(9), 2999–3019 (2009)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Dutkay D.E., Han D., Sun Q.: On the spectra of a Cantor measure. Adv. Math. 221(1), 251–276 (2009)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Dutkay D.E., Jorgensen P.E.T.: Fourier frequencies in affine iterated function systems. J. Funct. Anal. 247(1), 110–137 (2007)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Dutkay, D.E., Jorgensen P.E.T.: Fourier series on fractals: a parallel with wavelet theory. In: Radon Transforms, Geometry, and Wavelets. Contemporary Mathematics, vol. 464, pp. 75–101. American Mathematical Society, Providence (2008)Google Scholar
  7. 7.
    Dutkay D.E., Jorgensen P.E.T.: Duality questions for operators, spectrum and measures. Acta Appl. Math. 108(3), 515–528 (2009)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Dutkay, D.E., Jorgensen, P.E.T.: Fourier duality for fractal measures with affine scales. (2009)Google Scholar
  9. 9.
    Dutkay D.E., Jorgensen P.E.T.: Probability and Fourier duality for affine iterated function systems. Acta Appl. Math. 107(1–3), 293–311 (2009)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Dysman M.: Fractal dimensions for repellers of maps with holes. J. Stat. Phys. 120(3–4), 479–509 (2005)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Falconer K., Miao J.: Random subsets of self-affine fractals. Mathematika 56(1), 61–76 (2010)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Gentile, G.: Quasiperiodic motions in dynamical systems: review of a renormalization group approach. J. Math. Phys. 51(1):015207, 34 (2010)Google Scholar
  13. 13.
    Hutchinson J.E.: Fractals and self-similarity. Indiana Univ. Math. J 30(5), 713–747 (1981)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Ionescu M., Pearse E.P.J., Rogers L.G., Ruan H.-J., Strichartz R.S.: The resolvent kernel for PCF self-similar fractals. Trans. Am. Math. Soc 362(8), 4451–4479 (2010)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Iosevich A., Katz N., Pedersen S.: Fourier bases and a distance problem of Erdös. Math. Res. Lett. 6(2), 251–255 (1999)MathSciNetMATHGoogle Scholar
  16. 16.
    Iosevich A, Pedersen S.: Spectral and tiling properties of the unit cube. Internat. Math. Res. Notices 16, 819–828 (1998)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Jorgensen P.E.T.: Analysis and Probability: Wavelets, Signals, Fractals. Graduate Texts in Mathematics, vol. 234. Springer, New York (2006)Google Scholar
  18. 18.
    Jorgensen, P.E.T., Kornelson, K., Shuman K.: Orthogonal exponentials for Bernoulli iterated function systems. In: Representations, Wavelets, and Frames. Appl. Numer. Harmon. Anal, pp. 217–237. Birkhäuser, Boston (2008)Google Scholar
  19. 19.
    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)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Jorgensen P.E.T., Pedersen S.: Dense analytic subspaces in fractal L2-spaces. J. Anal. Math. 75, 185–228 (1998)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Jorgensen, P.E.T., Pedersen, S.: Orthogonal harmonic analysis of fractal measures. Electron. Res. Announc. Am. Math. Soc. 4:35–42 (1998, electronic)Google Scholar
  22. 22.
    Jorgensen P.E.T., Pedersen S.: Spectral pairs in Cartesian coordinates. J. Fourier Anal. Appl. 5(4), 285–302 (1999)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Katznelson Y.: An Introduction to Harmonic Analysis, 3rd edn. Cambridge Mathematical Library. Cambridge University Press, Cambridge (2004)MATHGoogle Scholar
  24. 24.
    Kukavica I.: The fractal dimension of the singular set for solutions of the Navier–Stokes system. Nonlinearity 22(12), 2889–2900 (2009)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Łaba I., Wang Y.: Some properties of spectral measures. Appl. Comput. Harmon. Anal. 20(1), 149–157 (2006)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Lagarias C.J., Reeds J.A., Wang Y.: Orthonormal bases of exponentials for the n-cube. Duke Math. J. 103(1), 25–37 (2000)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    O’Malley R.E. Jr, Kirkinis E.: A combined renormalization group-multiple scale method for singularly perturbed problems. Stud. Appl. Math. 124(4), 383–410 (2010)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Pedersen S.: Spectral sets whose spectrum is a lattice with a base. J. Funct. Anal. 141(2), 496–509 (1996)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Pedersen, S.: On the dual spectral set conjecture. In: Current Trends in Operator Theory and its Applications. Operator Theory Advances and Applications, vol. 149, pp. 487–491. Birkhäuser, Basel (2004)Google Scholar
  30. 30.
    Richey M.: The evolution of Markov chain Monte Carlo methods. Am. Math. Mon. 117(5), 383–413 (2010)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Rudin W.: Real and complex analysis, 3rd edn. McGraw-Hill Book., Co., New York (1987)MATHGoogle Scholar
  32. 32.
    Rudin, W.: Fourier analysis on groups. Wiley Classics Library. Wiley Inc., New York. Reprint of the 1962 original, A Wiley-Interscience Publication (1990)Google Scholar
  33. 33.
    Strichartz R.S.: Waves are recurrent on noncompact fractals. J. Fourier Anal. Appl. 16(1), 148–154 (2010)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Taylor, J.: The cohomology of the spectrum of a measure algebra. Acta Mathematica. 126, 195–225 (1971) doi:10.1007/BF02392031 Google Scholar
  35. 35.
    Wang X.Y.: Fractal dimensions of a class of random Moran sets. J. Math. (Wuhan) 29(3), 339–342 (2009)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Universität Bremen, Fachbereich 3, Mathematik und InformatikBremenGermany
  2. 2.Department of MathematicsThe University of IowaIowa CityUSA

Personalised recommendations