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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Infinite convolution products and refinable distributions on Lie groups

Author: Wayne Lawton
Journal: Trans. Amer. Math. Soc. 352 (2000), 2913-2936
MSC (1991): Primary 41A15, 41A58, 42C05, 42C15, 43A05, 43A15
Published electronically: March 2, 2000
MathSciNet review: 1638258
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Sufficient conditions for the convergence in distribution of an infinite convolution product $\mu_1*\mu_2*\ldots$ of measures on a connected Lie group $\mathcal G$ with respect to left invariant Haar measure are derived. These conditions are used to construct distributions $\phi$ that satisfy $T\phi = \phi$where $T$ is a refinement operator constructed from a measure $\mu$and a dilation automorphism $A$. The existence of $A$ implies $\mathcal G$ is nilpotent and simply connected and the exponential map is an analytic homeomorphism. Furthermore, there exists a unique minimal compact subset $\mathcal K \subset \mathcal G$such that for any open set $\mathcal U$ containing $\mathcal K,$ and for any distribution $f$ on $\mathcal G$ with compact support, there exists an integer $n(\mathcal U,f)$ such that $n \geq n(\mathcal U,f)$implies $\hbox{supp}(T^{n}f) \subset\mathcal U.$If $\mu$ is supported on an $A$-invariant uniform subgroup $\Gamma,$ then $T$ is related, by an intertwining operator, to a transition operator $W$ on $\mathbb C(\Gamma).$ Necessary and sufficient conditions for $T^{n}f$ to converge to $\phi \in L^{2}$, and for the $\Gamma$-translates of $\phi$ to be orthogonal or to form a Riesz basis, are characterized in terms of the spectrum of the restriction of $W$ to functions supported on $\Omega := \mathcal K \mathcal K^{-1} \cap \Gamma.$

References [Enhancements On Off] (What's this?)

  • 1. L. Bagget, A. Carey, W. Moran and P. Ohring, General existence theorems for orthonormal wavelets, an abstract approach, publications of the Research Institute of Mathematical Sciences, Kyoto University, #1, 31 (1995), 95-111. MR 96c:42060
  • 2. G. Birkhoff, A note on topological groups, Compositio Mathematica, 3 (1936), 427-430.
  • 3. N. Bourbaki, Groupes et algèbres de Lie. Chapitre I. Elements de Mathematique. Fasciule XXVI, Hermann, Paris, 1971. MR 42:6159
  • 4. A. S. Cavaretta, W. Dahmen and C. A. Micchelli, Stationary subdivision, Memoirs of the American Mathematical Society, 93 (1991), 1-186. MR 92h:65017
  • 5. A. Cohen, Ondelettes, analyses multiresolutions et traitement numerique du signal, PhD thesis, Universite Paris IX, Dauphine, 1990.
  • 6. A. Cohen and I. Daubechies, A stability criterion for biorthogonal wavelet bases and their subband coding scheme, Duke Mathematical Journal, 68 (1992), 313-335. MR 94b:94005
  • 7. A. Cohen, I. Daubechies and J. C. Feauveau, Biorthogonal basis of compactly supported wavelets, Communications on Pure and Applied Mathematics, 45 (1992), 485-560. MR 93e:42044
  • 8. L. Cornwin and F. P. Greenleaf, F. P., Representations of Nilpotent Lie Groups and their Applications, Cambridge University Press, 1990. MR 92b:22007
  • 9. X. Dai and D. R. Larson, Wandering vectors for unitary systems and orthonormal wavelets, to appear in Memoirs of the American Mathematical Society. MR 98m:47067
  • 10. I. Daubechies, Orthonormal bases of compactly supported wavelets, Communications on Pure and Applied Mathematics, 41 (1988), 909-996. MR 90m:42039
  • 11. I. Daubechies, Ten lectures on wavelets, CBMF Conference Series in Applied Mathematics, 61, SIAM, Philadephia, 1992. MR 93e:42045
  • 12. G. Deslauriers and S. Dubuc, Interpolation dyadique, in Fractals, Dimensions Non Entiérs et Applications (edited by G. Cherbit), Masson, Paris, 1987, pp. 44-45.
  • 13. J. Dixmier and W. G. Lister, Derivations of nilpotent Lie algebras, Proceedings of the American Mathematical Society, 8 (1957), 155-158. MR 18:659a
  • 14. J. L. Dyer, A nilpotent Lie algebra with nilpotent automorphism group, Bulletin of the American Mathematical Society, 76 (1970), 52-56. MR 40:2789
  • 15. N. Dyn, J. A. Gregory and D. Levin, Analysis of uniform binary subdivision schemes for curve design, Constructive Approximation, 7 (1991), 127-147. MR 92d:65027
  • 16. T. Eirola, Sobolev characterization of solutions of dilation equations, SIAM Journal of Mathematical Analysis, 23 (1992), 1015-1030. MR 93f:42056
  • 17. G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Arkiv fur Mathematik, 13 (1975), 161-207. MR 58:13215
  • 18. G. B. Folland, Harmonic Analysis in Phase Space, Princeton University Press, Princeton, New Jersey, 1989.
  • 19. S. A. Gall, Linear Analysis and Representation Theory, Springer-Verlag, New York, 1973.
  • 20. A. Grothendieck, Produits tensoriels et espaces nucléaires, Memoirs of the American Mathmatical Society, 16 (1955).
  • 21. A. Haar, Zur Theorie der orthogonalen Funktionensysteme, Mathematische Annalen, 69 (1910), 331-371.
  • 22. S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, New York, 1962. MR 26:2986
  • 23. P. N. Heller and R. O. Wells, Jr., The spectral theory of multiresolution operators and applications, in Wavelets: Theory, Algorithms, and Applications, (edited by C. K. Chui, L. Montefusco, L. Puccio), Academic Press, 1994, pp. 13-31. MR 96a:42046
  • 24. L. Hervé, Construction et regularite des fonctions d' echelle, SIAM J. Math. Anal., 5 (1995), 26.
  • 25. R. Hermann, Lie Groups for Physicists, Benjamin, New York, 1966. MR 35:4327
  • 26. J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1972. MR 48:2197
  • 27. N. Jacobson, A note on automorphisms and derivations of Lie algebras, Proceedings of the American Mathematical Society, 6 (1995), 281-283. MR 16:897e
  • 28. N. Jacobson, Lie Algebras, Wiley Interscience, New York, 1962. MR 26:1345
  • 29. R. Q. Jia, Subdivision Schemes in $L_p$ space, Advances in Computational Mathematics, 3 (1995), 309-341. MR 96d:65028
  • 30. R. Q. Jia and Z. Shen, Multiresolution and wavelets, Procedings Edinburgh Mathematical Society, 37 (1994), 271-300. MR 95h:42035
  • 31. S. Kakutani, Über die Metrization der Topologischen Gruppen, Proceedings of the Imperial Academay of Tokyo, 12 (1936), 82-84.
  • 32. W. Lawton, Tight frames of compactly supported affine wavelets, Journal of Mathematical Physics, #8, 31 (1990), 1898-1901. MR 92a:81068
  • 33. W. Lawton, Necessary and sufficient conditions for constructing orthonormal wavelet bases, Journal of Mathematical Physics, #1, 32 (1991), 57-61. MR 91m:81100
  • 34. W. Lawton, Multiresolution properties of the wavelet-Galerkin operator, Journal of Mathematical Physics, #6, 32 (1991), 1440-1443. MR 92f:42038
  • 35. W. Lawton and H. Resnikoff, Multidimensional Wavelet Bases, Technical Report, AWARE, Inc., Bedford, Massachusettes, 1991.
  • 36. W. Lawton, S. L. Lee and Z. Shen, Stability and orthonormality of multivariate refinable functions, SIAM Journal of Mathematical Analysis, #4, 28 (1997), 999-1014. MR 98d:41027
  • 37. W. Lawton, S. L. Lee and Z. Shen, Convergence of multidimensional cascade algorithm, Numerische Mathematik, 78 (1998), 427-438. MR 98k:41027
  • 38. P. G. Lemarie, Base d'ondelettes sur les groups de Lie stratifies, Bulletin Society Math. France, 117 (1989), 211-232. MR 90j:42066
  • 39. R. Long and D. Chen, Biorthogonal wavelet bases on $\RR^d$, Appl. Comp. Harmonic Anal., 2 (1995), 230-242. MR 96j:42012
  • 40. A. Malcev, On a class of homogeneous spaces, Izvestia Akademia Nauk SSSR Ser. Mat., 13 (1942), 9-32, American Mathematical Society Translations, vol. 39, 1949. MR 10:507d
  • 41. K. Maurin, General Eigenfunction Expansions and Unitary Representations of Topological Groups, Polish Scientific Publishers, Warsaw, 1968. MR 40:645
  • 42. Y. Meyer, Wavelets and Operators, Cambridge University Press, Cambridge, 1992. MR 94f:42001
  • 43. P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1993. MR 94g:58260
  • 44. W. Rudin, Functional Analysis, McGraw Hill, New York, 1973. MR 51:1315
  • 45. W. Rudin, Real and Complex Analysis, McGraw Hill, New York, 1974. MR 49:8783
  • 46. H. Samelson, Notes on Lie Algebras, Springer-Verlag, New York, 1990. MR 91h:17006
  • 47. O. L. Sattinger and O. L. Weaver, Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics, Springer-Verlag, New York, 1986.
  • 48. Z. Shen, Refinable function vectors, SIAM Journal of Mathematical Analysis, 29 (1998), 235-250. MR 99d:41038
  • 49. I. J. Schoenberg, Contributions to the problem of approximation of equidistant data by analytic functions, Quarterly Applied Mathematics., 4 (1946), 112-141. MR 8:55d
  • 50. L. Schwartz, Theorie des distributions, Hermann, Paris, 1957.
  • 51. G. Strang and G. Fix, Fourier analysis of the finite element method in Ritz-Galerkin Theory, Studies in Applied Mathematics, 48 (1969), 265-273. MR 41:2944
  • 52. R. Strichartz, A Guide to Distribution Theory and Fourier Transforms, CRC Press, Ann Arbor, 1994. MR 95f:42001
  • 53. R. Strichartz, Self-similarity on nilpotent Lie groups, Contemporary Mathematics, 140 (1992), 123-157. MR 94e:43011
  • 54. M. E. Taylor, Noncommutative harmonic analysis, Mathematical Surveys, No. 22, American Mathematical Society, Providence, 1986. MR 88a:22021
  • 55. F. Treves, F. Topological Vector Spaces, Distributions, and Kernels, Academic Press, New York, 1967. MR 37:726
  • 56. V. S. Varadarajan, Lie Groups, Lie Algebras, and their Representations, Springer-Verlag, New York, 1984. MR 85e:22001
  • 57. L. F. Villemoes, Energy moments in time and frequency for two-scale difference equation solutions and wavelets, SIAM Journal of Mathematical Analysis, 23 (1992), 1519-1543. MR 94c:39002
  • 58. K. Yoshida, Functional Analysis, Springer, New York, 1980.

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Additional Information

Wayne Lawton
Affiliation: Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543

Keywords: Lie group, distribution, enveloping algebra, dilation, refinement operator, cascade sequence, transition operator, condition E, Riesz basis
Received by editor(s): March 10, 1997
Received by editor(s) in revised form: April 9, 1998
Published electronically: March 2, 2000
Additional Notes: Research supported in part by the NUS Wavelets Program funded by the National Science and Technology Board and the Ministry of Education, Republic of Singapore.
Article copyright: © Copyright 2000 American Mathematical Society

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