Orthogonal harmonic analysis of fractal measures
Authors:
Palle E. T. Jorgensen and Steen Pedersen
Journal:
Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 3542
MSC (1991):
Primary 28A75, 42B10, 42C05; Secondary 47C05, 46L55
Published electronically:
May 5, 1998
MathSciNet review:
1618687
Fulltext 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.
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 [Fri87]
 J. Friedrich, On first order partial differential operators on bounded regions of the plane, Math. Nachr. 131 (1987), 3347. MR 88j:47062
 [Fug74]
 B. Fuglede, Commuting selfadjoint partial differential operators and a group theoretic problem, J. Funct. Anal. 16 (1974), 101121. MR 57:10500
 [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]
 P. E. T. Jorgensen, Spectral theory of finite volume domains in , Adv. Math. 44 (1982), 105120. MR 84k:47024
 [JP92]
 P. E. T. Jorgensen and S. Pedersen, Spectral theory for Borel sets in of finite measure, J. Funct. Anal. 107 (1992), 72104. MR 93k:47005
 [JP94]
 P. E. T. Jorgensen and S. Pedersen, Harmonic analysis and fractal limitmeasures induced by representations of a certain algebra, J. Funct. Anal. 125 (1994), 90110. MR 95i:47067
 [JP96]
 P. E. T. Jorgensen and S. Pedersen, Harmonic analysis of fractal measures, Constr. Approx. 12 (1996), 130. MR 97c:46091
 [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]
 J. Kigami and M. Lapidus, Weyl's problem for the spectral distribution of Laplacians on p.c.f. selfsimilar sets, Commun. Math. Phys. 158 (1993), 93125. MR 94m:58225
 [KL96]
 M. N. Kolountzakis and J. C. Lagarias, Structure of tilings of the line by a function, Duke Math. J. 82 (1996), 653678. MR 97d:11124
 [LRW98]
 J. C. Lagarias, J. A. Reed, and Y. Wang, Orthonormal bases of exponentials for the cube, preprint, 1998.
 [LW97]
 J. C. Lagarias and Y. Wang, Spectral sets and factorizations of finite Abelian groups, J. Funct. Anal. 145 (1997), 7398. MR 98b:47011b
 [Ped87]
 S. Pedersen, Spectral theory of commuting selfadjoint partial differential operators, J. Funct. Anal. 73 (1987), 122134. MR 89m:35163
 [Ped96]
 S. Pedersen, Spectral sets whose spectrum is a lattice with a base, J. Funct. Anal. 141 (1996), 496509. MR 98b:47011a
 [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]
 R. S. Strichartz, Selfsimilarity in harmonic analysis, J. Fourier Anal. Appl. 1 (1994), 137. MR 96c:42002
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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:
http://dx.doi.org/10.1090/S1079676298000444
PII:
S 10796762(98)000444
Keywords:
Spectral pair,
tiling,
Fourier basis,
selfsimilar 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
