An algebraic approach to multiresolution analysis

Author:
Richard Foote

Journal:
Trans. Amer. Math. Soc. **357** (2005), 5031-5050

MSC (2000):
Primary 20C99; Secondary 42C40

Published electronically:
March 18, 2005

MathSciNet review:
2165396

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The notion of a weak multiresolution analysis is defined over an arbitrary field in terms of cyclic modules for a certain affine group ring. In this setting the basic properties of weak multiresolution analyses are established, including characterizations of their submodules and quotient modules, the existence and uniqueness of reduced scaling equations, and the existence of wavelet bases. These results yield some standard facts on classical multiresolution analyses over the reals as special cases, but provide a different perspective by not relying on orthogonality or topology. Connections with other areas of algebra and possible further directions are mentioned.

**[1]**J. Benedetto and R. Benedetto,*A wavelet theory for local fields and related groups*, submitted for publication.**[2]**R. Benedetto,*Examples of wavelets over local fields*, Contemporary Mathematics (2003).**[3]**R. Bernanardini and J. Kovacevic,*Local orthogonal bases I and II*, Multidim. Syst. Signal Process.**7**(1996), 331-370 and 371-400.**[4]**Ola Bratteli and Palle E. T. Jorgensen,*Isometries, shifts, Cuntz algebras and multiresolution wavelet analysis of scale 𝑁*, Integral Equations Operator Theory**28**(1997), no. 4, 382–443. MR**1465320**, 10.1007/BF01309155**[5]**Gregory S. Chirikjian and Alexander B. Kyatkin,*Engineering applications of noncommutative harmonic analysis*, CRC Press, Boca Raton, FL, 2001. With emphasis on rotation and motion groups. MR**1885369****[6]**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****[7]**D. Dummit and R. Foote,*Abstract Algebra, Third Edition*, John Wiley & Sons, Inc, NJ, 2004.**[8]**K. Flornes, A. Grossmann, M. Holschneider, and B. Torrésani,*Wavelets on discrete fields*, Appl. Comput. Harmon. Anal.**1**(1994), no. 2, 137–146. MR**1310638**, 10.1006/acha.1994.1001**[9]**Richard Foote, Gagan Mirchandani, Daniel N. Rockmore, Dennis Healy, and Tim Olson,*A wreath product group approach to signal and image processing. I. Multiresolution analysis*, IEEE Trans. Signal Process.**48**(2000), no. 1, 102–132. MR**1736279**, 10.1109/78.815483**[10]**David Goss,*Basic structures of function field arithmetic*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 35, Springer-Verlag, Berlin, 1996. MR**1423131****[11]**Alex Grossmann and Thierry Paul,*Wave functions on subgroups of the group of affine canonical transformations*, Resonances—models and phenomena (Bielefeld, 1984) Lecture Notes in Phys., vol. 211, Springer, Berlin, 1984, pp. 128–138. MR**777335**, 10.1007/3-540-13880-3_69**[12]**A. Grossmann and J. Morlet,*Decomposition of Hardy functions into square integrable wavelets of constant shape*, SIAM J. Math. Anal.**15**(1984), no. 4, 723–736. MR**747432**, 10.1137/0515056**[13]**A. Grossmann, J. Morlet, and T. Paul,*Transforms associated to square integrable group representations. I. General results*, J. Math. Phys.**26**(1985), no. 10, 2473–2479. MR**803788**, 10.1063/1.526761

A. Grossmann, J. Morlet, and T. Paul,*Transforms associated to square integrable group representations. II. Examples*, Ann. Inst. H. Poincaré Phys. Théor.**45**(1986), no. 3, 293–309 (English, with French summary). MR**868528****[14]**Michael Kirby,*Geometric data analysis*, Wiley-Interscience [John Wiley & Sons], New York, 2001. An empirical approach to dimensionality reduction and the study of patterns. MR**1874451****[15]**Alexander B. Kyatkin and Gregory S. Chirikjian,*Algorithms for fast convolutions on motion groups*, Appl. Comput. Harmon. Anal.**9**(2000), no. 2, 220–241. MR**1777127**, 10.1006/acha.2000.0321**[16]**Stephane G. Mallat,*Multiresolution approximations and wavelet orthonormal bases of 𝐿²(𝑅)*, Trans. Amer. Math. Soc.**315**(1989), no. 1, 69–87. MR**1008470**, 10.1090/S0002-9947-1989-1008470-5**[17]**D. Maslen,*Sampling of functions and sections for compact groups*, Modern Signal Processing, D. Rockmore and D. Healy, eds., Cambridge University Press, Cambridge, to appear.**[18]**David K. Maslen,*Efficient computation of Fourier transforms on compact groups*, J. Fourier Anal. Appl.**4**(1998), no. 1, 19–52. MR**1650948**, 10.1007/BF02475926**[19]**David K. Maslen and Daniel N. Rockmore,*The Cooley-Tukey FFT and group theory*, Notices Amer. Math. Soc.**48**(2001), no. 10, 1151–1160. MR**1861656****[20]**David K. Maslen and Daniel N. Rockmore,*Generalized FFTs—a survey of some recent results*, Groups and computation, II (New Brunswick, NJ, 1995) DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 28, Amer. Math. Soc., Providence, RI, 1997, pp. 183–237. MR**1444138****[21]**Gagan Mirchandani, Richard Foote, Daniel N. Rockmore, Dennis Healy, and Tim Olson,*A wreath product group approach to signal and image processing. II. Convolution, correlation, and applications*, IEEE Trans. Signal Process.**48**(2000), no. 3, 749–767. MR**1765933**, 10.1109/78.824670**[22]**Murad Özaydin and Tomasz Przebinda,*Platonic orthonormal wavelets*, Appl. Comput. Harmon. Anal.**4**(1997), no. 4, 351–365. MR**1474094**, 10.1006/acha.1997.0212**[23]**Daniel N. Rockmore,*Some applications of generalized FFTs*, Groups and computation, II (New Brunswick, NJ, 1995) DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 28, Amer. Math. Soc., Providence, RI, 1997, pp. 329–369. MR**1444144****[24]**Allan J. Silberger,*Introduction to harmonic analysis on reductive 𝑝-adic groups*, Mathematical Notes, vol. 23, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1979. Based on lectures by Harish-Chandra at the Institute for Advanced Study, 1971–1973. MR**544991****[25]**Gilbert Strang and Truong Nguyen,*Wavelets and filter banks*, Wellesley-Cambridge Press, Wellesley, MA, 1996. MR**1411910****[26]**B. Yazici,*Group invariant methods in signal processing*, Proc. Conf. on Information Science and Systems, Johns Hopkins Univ., MD, 1997.**[27]**B. Yazici,*Stocastic deconvolution over groups*, IEEE Trans. in Info. Th., to appear.

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Additional Information

**Richard Foote**

Affiliation:
Department of Mathematics and Statistics, University of Vermont, Burlington, Vermont 05405

Email:
foote@math.uvm.edu

DOI:
http://dx.doi.org/10.1090/S0002-9947-05-03656-1

Keywords:
Multiresolution analysis,
wavelets

Received by editor(s):
March 7, 2003

Received by editor(s) in revised form:
January 14, 2004, and March 5, 2004

Published electronically:
March 18, 2005

Additional Notes:
This work was partially supported by an AFOSR/NM grant

Article copyright:
© Copyright 2005
American Mathematical Society