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Low-pass filters and representations of the Baumslag Solitar group
Author:
Dorin Ervin Dutkay
Journal:
Trans. Amer. Math. Soc. 358 (2006), 5271-5291
MSC (2000):
Primary 42C40, 28A78, 46L45, 28D05, 22D25
Posted:
July 21, 2006
MathSciNet review:
2238916
Full-text PDF Free Access
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Abstract: We analyze representations of the Baumslag Solitar group  that admit wavelets and show how such representations can be constructed from a given low-pass filter. We describe the direct integral decomposition for some examples and derive from it a general criterion for the existence of solutions for scaling equations. As another application, we construct a Fourier transform for some Hausdorff measures.
- [BDP]
Stefan
Bildea, Dorin
Ervin Dutkay, and Gabriel
Picioroaga, MRA super-wavelets, New York J. Math.
11 (2005), 1–19. MR 2154344
(2006b:42048)
- [Dau92]
Ingrid
Daubechies, Ten lectures on wavelets, CBMS-NSF Regional
Conference Series in Applied Mathematics, vol. 61, Society for
Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1162107
(93e:42045)
- [Dut2]
Dorin
E. Dutkay, The wavelet Galerkin operator, J. Operator Theory
51 (2004), no. 1, 49–70. MR 2055804
(2005c:47041)
- [Dut3]
Dorin
Ervin Dutkay, Positive definite maps, representations and
frames, Rev. Math. Phys. 16 (2004), no. 4,
451–477. MR 2065233
(2005e:42115), http://dx.doi.org/10.1142/S0129055X04002047
- [DutJo]
D.E. Dutkay, P.E.T. Jorgensen, Wavelets on fractals, to appear in Revista Matematica Iberoamericana.
- [Fol]
Gerald
B. Folland, A course in abstract harmonic analysis, Studies in
Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1397028
(98c:43001)
- [GS]
Gernot
Greschonig and Klaus
Schmidt, Ergodic decomposition of quasi-invariant probability
measures. part 2, Colloq. Math. 84/85 (2000),
no. part 2, 495–514. Dedicated to the memory of Anzelm Iwanik.
MR
1784210 (2001i:28021)
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F. Gundy, Low-pass filters, martingales, and multiresolution
analyses, Appl. Comput. Harmon. Anal. 9 (2000),
no. 2, 204–219. MR 1777126
(2002k:94013), http://dx.doi.org/10.1006/acha.2000.0320
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E. T. Jorgensen, Ruelle operators: functions which are harmonic
with respect to a transfer operator, Mem. Amer. Math. Soc.
152 (2001), no. 720, viii+60. MR 1837681
(2002c:46117)
- [Jor04]
P.E.T. Jorgensen, Processes, wavelets and random walks on branches, preprint.
- [LPT]
Lek-Heng
Lim, Judith
A. Packer, and Keith
F. Taylor, A direct integral decomposition of the
wavelet representation, Proc. Amer. Math.
Soc. 129 (2001), no. 10, 3057–3067 (electronic). MR 1840112
(2002c:47146), http://dx.doi.org/10.1090/S0002-9939-01-05928-7
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Florian
Martin and Alain
Valette, Markov operators on the solvable Baumslag-Solitar
groups, Experiment. Math. 9 (2000), no. 2,
291–300. MR 1780213
(2002a:37005)
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Peter
Walters, An introduction to ergodic theory, Graduate Texts in
Mathematics, vol. 79, Springer-Verlag, New York, 1982. MR 648108
(84e:28017)
- [Web]
E. Weber, Discrete wavelet transforms and
 -admissible group representations, preliminary version.
- [BDP]
- S. Bildea, D.E. Dutkay, G. Picioroaga, MRA Super-wavelets, New York J. Math., 11 (2005), 1-19. MR 2154344 (2006b:42048)
- [Dau92]
- I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conf. Ser. in Appl. Math., vol. 61, Society for Industrial and Applied Mathematics, Philadelphia, 1992. MR 1162107 (93e:42045)
- [Dut2]
- D.E. Dutkay, The wavelet Galerkin operator, J. Operator Theory, 51(2004), 49-70. MR 2055804 (2005c:47041)
- [Dut3]
- D.E. Dutkay, Positive definite maps, representations and frames, Reviews in Mathematical Physics, vol. 16, No. 4 (2004), 1-27. MR 2065233 (2005e:42115)
- [DutJo]
- D.E. Dutkay, P.E.T. Jorgensen, Wavelets on fractals, to appear in Revista Matematica Iberoamericana.
- [Fol]
- G. Folland, A Course in Abstract Harmonic Analysis, CRC Press, 1995. MR 1397028 (98c:43001)
- [GS]
- G. Greschonig, K. Schmidt, Ergodic decomposition of quasi-invariant probability measures, Colloq. Math. 84/85 (2000), part 2, 495-514. MR 1784210 (2001i:28021)
- [Gun00]
- R.F. Gundy, Low-pass flters, martingales, and multiresolution analyses, Appl. Comput. Harmon. Anal., 9 (2000), 204-219. MR 1777126 (2002k:94013)
- [Jor01]
- P.E.T. Jorgensen, Ruelle operators: Functions which are harmonic with respect to a transfer operator, Mem. Amer. Math. Soc., 152, no. 720. MR 1837681 (2002c:46117)
- [Jor04]
- P.E.T. Jorgensen, Processes, wavelets and random walks on branches, preprint.
- [LPT]
- L.-H. Lim, J. Packer, K. Taylor, A direct integral decomposition of the wavelet representation, Proc. Amer. Soc., 129, No. 10, 3057-3067. MR 1840112 (2002c:47146)
- [MV00]
- F. Martin, A. Valette, Markov operators on the solvable Baumslag-Solitar groups, Experimental Mathematics, Vol. 9 (2000), No. 2. MR 1780213 (2002a:37005)
- [Wal]
- P. Walters, An introduction to ergodic theory, Springer-Verlag, New York, 1982. MR 0648108 (84e:28017)
- [Web]
- E. Weber, Discrete wavelet transforms and -admissible group representations, preliminary version.
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Additional Information
Dorin Ervin Dutkay
Affiliation:
Department of Mathematics, Hill Center-Busch Campus, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019
Email:
ddutkay@math.rutgers.edu
DOI:
http://dx.doi.org/10.1090/S0002-9947-06-04230-9
PII:
S 0002-9947(06)04230-9
Keywords:
Wavelet,
representation,
low-pass filter,
scaling function,
Baumslag Solitar group,
solenoid,
decomposition,
ergodic action,
fractal
Received by editor(s):
July 21, 2004
Posted:
July 21, 2006
Additional Notes:
This work was supported in part by NSF grant DMS0457491
Article copyright:
© Copyright 2006 American Mathematical Society
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