Orthogonal harmonic analysis of fractal measures
Authors:
Palle E. T. Jorgensen and Steen Pedersen
Journal:
Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 35-42
MSC (1991):
Primary 28A75, 42B10, 42C05; Secondary 47C05, 46L55
DOI:
https://doi.org/10.1090/S1079-6762-98-00044-4
Published electronically:
May 5, 1998
MathSciNet review:
1618687
Full-text PDF Free Access
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Additional Information
Abstract: We show that certain iteration systems lead to fractal measures admitting an exact orthogonal harmonic analysis.
- G. Björck and B. Saffari, New classes of finite unimodular sequences with unimodular Fourier transforms. Circulant Hadamard matrices with complex entries, C. R. Acad. Sci. Paris, Série 1, 320 (1995), 319–324.
- Jürgen Friedrich, On first order partial differential operators on bounded regions of the plane, Math. Nachr. 131 (1987), 33–47. MR 908797, DOI https://doi.org/10.1002/mana.19871310104
- Bent Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem, J. Functional Analysis 16 (1974), 101–121. MR 0470754, DOI https://doi.org/10.1016/0022-1236%2874%2990072-x
- U. Haagerup, Orthogonal maximal Abelian *-subalgebras of the $n\times n$ matrices and cyclic $n$-roots, preprint, 1995, 29 pp.
- A. Iosevich and S. Pedersen, Spectral and tiling properties of the unit cube, preprint, 1998.
- Palle E. T. Jørgensen, Spectral theory of finite volume domains in ${\bf R}^{n}$, Adv. in Math. 44 (1982), no. 2, 105–120. MR 658536, DOI https://doi.org/10.1016/0001-8708%2882%2990001-9
- Palle E. T. Jorgensen and Steen Pedersen, Spectral theory for Borel sets in ${\bf R}^n$ of finite measure, J. Funct. Anal. 107 (1992), no. 1, 72–104. MR 1165867, DOI https://doi.org/10.1016/0022-1236%2892%2990101-N
- Palle E. T. Jorgensen and Steen Pedersen, Harmonic analysis and fractal limit-measures induced by representations of a certain $C^*$-algebra, J. Funct. Anal. 125 (1994), no. 1, 90–110. MR 1297015, DOI https://doi.org/10.1006/jfan.1994.1118
- P. E. T. Jorgensen and S. Pedersen, Harmonic analysis of fractal measures, Constr. Approx. 12 (1996), no. 1, 1–30. MR 1389918, DOI https://doi.org/10.1007/s003659900001
- P. E. T. Jorgensen and S. Pedersen, Spectral pairs in Cartesian coordinates, J. Fourier Anal. Appl., to appear.
- P. E. T. Jorgensen and S. Pedersen, Dense analytic subspaces in fractal $L^2$-spaces, J. Anal. Math., to appear.
- Jun Kigami and Michel L. Lapidus, Weyl’s problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals, Comm. Math. Phys. 158 (1993), no. 1, 93–125. MR 1243717
- Mihail N. Kolountzakis and Jeffrey C. Lagarias, Structure of tilings of the line by a function, Duke Math. J. 82 (1996), no. 3, 653–678. MR 1387688, DOI https://doi.org/10.1215/S0012-7094-96-08227-7
- J. C. Lagarias, J. A. Reed, and Y. Wang, Orthonormal bases of exponentials for the $n$-cube, preprint, 1998.
- Steen Pedersen, Spectral sets whose spectrum is a lattice with a base, J. Funct. Anal. 141 (1996), no. 2, 496–509. MR 1418517, DOI https://doi.org/10.1006/jfan.1996.0139
- Steen Pedersen, Spectral theory of commuting selfadjoint partial differential operators, J. Funct. Anal. 73 (1987), no. 1, 122–134. MR 890659, DOI https://doi.org/10.1016/0022-1236%2887%2990061-9
- Steen Pedersen, Spectral sets whose spectrum is a lattice with a base, J. Funct. Anal. 141 (1996), no. 2, 496–509. MR 1418517, DOI https://doi.org/10.1006/jfan.1996.0139
- S. Pedersen, Fourier series and geometry, preprint, 1997.
- S. Pedersen and Y. Wang, Spectral sets, translation tiles and characteristic polynomials, preprint, 1997.
- Robert S. Strichartz, Self-similarity in harmonic analysis, J. Fourier Anal. Appl. 1 (1994), no. 1, 1–37. MR 1307067, DOI https://doi.org/10.1007/s00041-001-4001-z
- G. Björck and B. Saffari, New classes of finite unimodular sequences with unimodular Fourier transforms. Circulant Hadamard matrices with complex entries, C. R. Acad. Sci. Paris, Série 1, 320 (1995), 319–324.
- J. Friedrich, On first order partial differential operators on bounded regions of the plane, Math. Nachr. 131 (1987), 33–47.
- B. Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem, J. Funct. Anal. 16 (1974), 101–121.
- U. Haagerup, Orthogonal maximal Abelian *-subalgebras of the $n\times n$ matrices and cyclic $n$-roots, preprint, 1995, 29 pp.
- A. Iosevich and S. Pedersen, Spectral and tiling properties of the unit cube, preprint, 1998.
- P. E. T. Jorgensen, Spectral theory of finite volume domains in $\mathbb {R}^{n}$, Adv. Math. 44 (1982), 105–120.
- P. E. T. Jorgensen and S. Pedersen, Spectral theory for Borel sets in $\mathbb {R}^{n}$ of finite measure, J. Funct. Anal. 107 (1992), 72–104.
- P. E. T. Jorgensen and S. Pedersen, Harmonic analysis and fractal limit-measures induced by representations of a certain $C^{*}$-algebra, J. Funct. Anal. 125 (1994), 90–110.
- P. E. T. Jorgensen and S. Pedersen, Harmonic analysis of fractal measures, Constr. Approx. 12 (1996), 1–30.
- P. E. T. Jorgensen and S. Pedersen, Spectral pairs in Cartesian coordinates, J. Fourier Anal. Appl., to appear.
- P. E. T. Jorgensen and S. Pedersen, Dense analytic subspaces in fractal $L^2$-spaces, J. Anal. Math., to appear.
- J. Kigami and M. Lapidus, Weyl’s problem for the spectral distribution of Laplacians on p.c.f. self-similar sets, Commun. Math. Phys. 158 (1993), 93–125.
- M. N. Kolountzakis and J. C. Lagarias, Structure of tilings of the line by a function, Duke Math. J. 82 (1996), 653–678.
- J. C. Lagarias, J. A. Reed, and Y. Wang, Orthonormal bases of exponentials for the $n$-cube, preprint, 1998.
- J. C. Lagarias and Y. Wang, Spectral sets and factorizations of finite Abelian groups, J. Funct. Anal. 145 (1997), 73–98.
- S. Pedersen, Spectral theory of commuting self-adjoint partial differential operators, J. Funct. Anal. 73 (1987), 122–134.
- S. Pedersen, Spectral sets whose spectrum is a lattice with a base, J. Funct. Anal. 141 (1996), 496–509.
- S. Pedersen, Fourier series and geometry, preprint, 1997.
- S. Pedersen and Y. Wang, Spectral sets, translation tiles and characteristic polynomials, preprint, 1997.
- R. S. Strichartz, Self-similarity in harmonic analysis, J. Fourier Anal. Appl. 1 (1994), 1–37.
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Additional Information
Palle E. T. Jorgensen
Affiliation:
Department of Mathematics, University of Iowa, Iowa City, IA 52242
MR Author ID:
95800
ORCID:
0000-0003-2681-5753
Email:
jorgen@math.uiowa.edu
Steen Pedersen
Affiliation:
Department of Mathematics, Wright State University, Dayton, OH 45435
MR Author ID:
247731
Email:
steen@math.wright.edu
Keywords:
Spectral pair,
tiling,
Fourier basis,
self-similar measure,
fractal,
affine iteration,
spectral resolution,
Hilbert space
Received by editor(s):
October 13, 1997
Published electronically:
May 5, 1998
Communicated by:
Yitzhak Katznelson
Article copyright:
© Copyright 1998
American Mathematical Society
