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

Published electronically:
May 5, 1998

MathSciNet review:
1618687

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that certain iteration systems lead to fractal measures admitting an exact orthogonal harmonic analysis.

**[BS95]**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. CMP**95:09****[Fri87]**Jürgen Friedrich,*On first order partial differential operators on bounded regions of the plane*, Math. Nachr.**131**(1987), 33–47. MR**908797**, 10.1002/mana.19871310104**[Fug74]**Bent Fuglede,*Commuting self-adjoint partial differential operators and a group theoretic problem*, J. Functional Analysis**16**(1974), 101–121. MR**0470754****[Haa95]**U. Haagerup,*Orthogonal maximal Abelian *-subalgebras of the matrices and cyclic -roots*, preprint, 1995, 29 pp.**[IP98]**A. Iosevich and S. Pedersen,*Spectral and tiling properties of the unit cube*, preprint, 1998.**[Jor82]**Palle E. T. Jørgensen,*Spectral theory of finite volume domains in 𝑅ⁿ*, Adv. in Math.**44**(1982), no. 2, 105–120. MR**658536**, 10.1016/0001-8708(82)90001-9**[JP92]**Palle E. T. Jorgensen and Steen Pedersen,*Spectral theory for Borel sets in 𝑅ⁿ of finite measure*, J. Funct. Anal.**107**(1992), no. 1, 72–104. MR**1165867**, 10.1016/0022-1236(92)90101-N**[JP94]**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**, 10.1006/jfan.1994.1118**[JP96]**P. E. T. Jorgensen and S. Pedersen,*Harmonic analysis of fractal measures*, Constr. Approx.**12**(1996), no. 1, 1–30. MR**1389918**, 10.1007/s003659900001**[JP97a]**P. E. T. Jorgensen and S. Pedersen,*Spectral pairs in Cartesian coordinates*, J. Fourier Anal. Appl., to appear.**[JP97b]**P. E. T. Jorgensen and S. Pedersen,*Dense analytic subspaces in fractal -spaces*, J. Anal. Math., to appear.**[KL93]**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****[KL96]**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**, 10.1215/S0012-7094-96-08227-7**[LRW98]**J. C. Lagarias, J. A. Reed, and Y. Wang,*Orthonormal bases of exponentials for the -cube*, preprint, 1998.**[LW97]**Jeffrey C. Lagarias and Yang Wang,*Spectral sets and factorizations of finite abelian groups*, J. Funct. Anal.**145**(1997), no. 1, 73–98. MR**1442160**, 10.1006/jfan.1996.3008**[Ped87]**Steen Pedersen,*Spectral theory of commuting selfadjoint partial differential operators*, J. Funct. Anal.**73**(1987), no. 1, 122–134. MR**890659**, 10.1016/0022-1236(87)90061-9**[Ped96]**Steen Pedersen,*Spectral sets whose spectrum is a lattice with a base*, J. Funct. Anal.**141**(1996), no. 2, 496–509. MR**1418517**, 10.1006/jfan.1996.0139**[Ped97]**S. Pedersen,*Fourier series and geometry*, preprint, 1997.**[PW97]**S. Pedersen and Y. Wang,*Spectral sets, translation tiles and characteristic polynomials*, preprint, 1997.**[Str94]**Robert S. Strichartz,*Self-similarity in harmonic analysis*, J. Fourier Anal. Appl.**1**(1994), no. 1, 1–37. MR**1307067**, 10.1007/s00041-001-4001-z

Retrieve articles in *Electronic Research Announcements of the American Mathematical Society*
with MSC (1991):
28A75,
42B10,
42C05,
47C05,
46L55

Retrieve articles in all journals with MSC (1991): 28A75, 42B10, 42C05, 47C05, 46L55

Additional Information

**Palle E. T. Jorgensen**

Affiliation:
Department of Mathematics, University of Iowa, Iowa City, IA 52242

Email:
jorgen@math.uiowa.edu

**Steen Pedersen**

Affiliation:
Department of Mathematics, Wright State University, Dayton, OH 45435

Email:
steen@math.wright.edu

DOI:
https://doi.org/10.1090/S1079-6762-98-00044-4

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