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

Published electronically:
July 21, 2006

MathSciNet review:
2238916

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We analyze representations of the Baumslag Solitar group

**[BDP]**Stefan Bildea, Dorin Ervin Dutkay, and Gabriel Picioroaga,*MRA super-wavelets*, New York J. Math.**11**(2005), 1–19. MR**2154344****[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****[Dut2]**Dorin E. Dutkay,*The wavelet Galerkin operator*, J. Operator Theory**51**(2004), no. 1, 49–70. MR**2055804****[Dut3]**Dorin Ervin Dutkay,*Positive definite maps, representations and frames*, Rev. Math. Phys.**16**(2004), no. 4, 451–477. MR**2065233**, 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****[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****[Gun00]**Richard F. Gundy,*Low-pass filters, martingales, and multiresolution analyses*, Appl. Comput. Harmon. Anal.**9**(2000), no. 2, 204–219. MR**1777126**, 10.1006/acha.2000.0320**[Jor01]**Palle 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**, 10.1090/memo/0720**[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**, 10.1090/S0002-9939-01-05928-7**[MV00]**Florian Martin and Alain Valette,*Markov operators on the solvable Baumslag-Solitar groups*, Experiment. Math.**9**(2000), no. 2, 291–300. MR**1780213****[Wal]**Peter Walters,*An introduction to ergodic theory*, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR**648108****[Web]**E. Weber,*Discrete wavelet transforms and -admissible group representations*, preliminary version.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
42C40,
28A78,
46L45,
28D05,
22D25

Retrieve articles in all journals with MSC (2000): 42C40, 28A78, 46L45, 28D05, 22D25

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

Keywords:
Wavelet,
representation,
low-pass filter,
scaling function,
Baumslag Solitar group,
solenoid,
decomposition,
ergodic action,
fractal

Received by editor(s):
July 21, 2004

Published electronically:
July 21, 2006

Additional Notes:
This work was supported in part by NSF grant DMS0457491

Article copyright:
© Copyright 2006
American Mathematical Society