Abstract
The measure supported on the Cantor-4 set constructed by Jorgensen–Pedersen is known to have a Fourier basis, i.e. that it possess a sequence of exponentials which form an orthonormal basis. We construct Fourier frames for this measure via a dilation theory type construction. We expand the Cantor-4 set to a two dimensional fractal which admits a representation of a Cuntz algebra. Using the action of this algebra, an orthonormal set is generated on the larger fractal, which is then projected onto the Cantor-4 set to produce a Fourier frame.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aldroubi, A.: Portraits of frames. Proc. Am. Math. Soc. 123(6), 1661–1668 (1995)
Dai, X.-R., He, X.-G., Lai, C.-K.: Spectral property of Cantor measures with consecutive digits. Adv. Math. 242, 187–208 (2013)
Duffin, R.J., Schaeffer, A.C.: A class of nonharmonic Fourier series. Trans. Am. Math. Soc. 72, 341–366 (1952)
Dutkay, D.E., Han, D., Sun, Q.: On the spectra of a Cantor measure. Adv. Math. 221(1), 251–276 (2009)
Dutkay, D.E., Han, D., Weber, E.: Bessel sequences of exponentials on fractal measures. J. Funct. Anal. 261(9), 2529–2539 (2011)
Dutkay, D.E., Han, D., Weber, E.: Continuous and discrete Fourier frames for fractal measures. Trans. Am. Math. Soc. 366(3), 1213–1235 (2014)
Dutkay, D.E., Picioroaga, G., Song, M.-S.: Orthonormal bases generated by Cuntz algebras. J. Math. Anal. Appl. 409(2), 1128–1139 (2014)
Han, D., Larson, D.R.: Frames, bases and group representations, Mem. Am. Math. Soc. 147 (2000), no. 697, x+94. MR 1686653 (2001a:47013)
Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30(5), 713–747 (1981)
Jorgensen, P.E.T., Pedersen, S.: Dense analytic subspaces in fractal \(L^2\)-spaces. J. Anal. Math. 75, 185–228 (1998)
Łaba, I., Wang, Y.: On spectral Cantor measures. J. Funct. Anal. 193(2), 409–420 (2002)
Li, J.-L.: \(\mu _{M, D}\)-Orthogonality and compatible pair. J. Funct. Anal. 244(2), 628–638 (2007)
Ortega-Cerdà, J., Seip, K.: Fourier frames, Ann. Math. 155(3), 789–806 (2002)
Strichartz, R.S.: Mock Fourier series and transforms associated with certain Cantor measures. J. Anal. Math. 81, 209–238 (2000)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Chris Heil.
Rights and permissions
About this article
Cite this article
Picioroaga, G., Weber, E.S. Fourier Frames for the Cantor-4 Set. J Fourier Anal Appl 23, 324–343 (2017). https://doi.org/10.1007/s00041-016-9471-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00041-016-9471-0