Abstract
We investigate some relations between number theory and spectral measures related to the harmonic analysis of a Cantor set. Specifically, we explore ways to determine when an odd natural number m generates a complete or incomplete Fourier basis for a Cantor-type measure with scale g.
Similar content being viewed by others
References
Dorin Ervin Dutkay and John Haussermann, Number theory problems from the harmonic analysis of a fractal, J. Number Theory, 159 (2016), 7–26.
Xin-Rong Dai, Xing-Gang He and Chun-Kit Lai, Spectral property of Cantor measures with consecutive digits, Adv. Math., 242 (2013), 187–208.
Dorin Ervin Dutkay and Palle E. T. Jorgensen, Iterated function systems, Ruelle operators, and invariant projective measures, Math. Comp., 75 (2006), 1931–1970 (electronic).
Dorin Ervin Dutkay and Palle E. T. Jorgensen, Fourier frequencies in affine iterated function systems, J. Funct. Anal., 247 (2007), 110–137.
Dorin Ervin Dutkay and Palle E. T. Jorgensen, Fourier duality for fractal measures with affine scales, Math. Comp., 81 (2012), 2253–2273.
John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713–747.
Palle E. T. Jorgensen, Keri A. Kornelson and Karen L. Shuman, An operator-fractal, Numer. Funct. Anal. Optim., 33 (2012), 1070–1094.
Palle E. T. Jorgensen, Keri A. Kornelson and Karen L. Shuman, Scalar spectral measures associated with an operator-fractal, J. Math. Phys., 55 (2014), 022103, 23 pp.
Palle E. T. Jorgensen, Keri A. Kornelson and Karen L. Shuman, Scaling by 5 on a 1 4 -Cantor measure, Rocky Mountain J. Math., 44 (2014), 1881–1901.
Palle E. T. Jorgensen and Steen Pedersen, Dense analytic subspaces in fractal L2- spaces, J. Anal. Math., 75 (1998), 185–228.
Jian-Lin Li, μM,D-orthogonality and compatible pair, J. Funct. Anal., 244 (2007), 628–638.
Izabella Laba and Yang Wang, On spectral Cantor measures, J. Funct. Anal., 193 (2002), 409–420.
Robert S. Strichartz, Mock Fourier series and transforms associated with certain Cantor measures, J. Anal. Math., 81 (2000), 209–238.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the memory of Jean-Pierre Kahane
This material is based upon work supported by the National Science Foundation under Award No. 1356233. This work was partially supported by a grant from the Simons Foundation (#228539 to Dorin Dutkay).
Rights and permissions
About this article
Cite this article
Dutkay, D.E., Kraus, I. Number Theoretic Considerations Related to the Scaling of Spectra of Cantor-Type Measures. Anal Math 44, 335–367 (2018). https://doi.org/10.1007/s10476-018-0505-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10476-018-0505-5