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.

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*Spectral pairs in Cartesian coordinates*, J. Fourier Anal. Appl., to appear.
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*Dense analytic subspaces in fractal $L^2$-spaces*, J. Anal. Math., to appear.
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*Orthonormal bases of exponentials for the $n$-cube*, preprint, 1998.
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*Spectral sets whose spectrum is a lattice with a base*, J. Funct. Anal. **141** (1996), no. 2, 496–509. MR **1418517**, DOI 10.1006/jfan.1996.0139
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*Fourier series and geometry*, preprint, 1997.
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- 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