# zbMATH — the first resource for mathematics

Infinite convolution products and refinable distributions on Lie groups. (English) Zbl 0940.43002
Summary: “Sufficient conditions for the convergence in distribution of an infinite convolution product $$\mu_1\ast\mu_2\ast\cdots$$ 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 supp $$(T^nf)\subset{\mathcal U}$$. 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 $${\mathbf{C}}(\Gamma)$$. 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 $$\Omega:={\mathcal K}{\mathcal K}^{-1}\cap\Gamma$$”.

##### MSC:
 43A05 Measures on groups and semigroups, etc. 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis 41A15 Spline approximation 41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series) 43A15 $$L^p$$-spaces and other function spaces on groups, semigroups, etc.
Full Text:
##### References:
 [1] Larry Baggett, Alan Carey, William Moran, and Peter Ohring, General existence theorems for orthonormal wavelets, an abstract approach, Publ. Res. Inst. Math. Sci. 31 (1995), no. 1, 95 – 111. · Zbl 0849.42019 [2] G. Birkhoff, A note on topological groups, Compositio Mathematica, 3 (1936), 427-430. · JFM 62.0434.03 [3] N. Bourbaki, Éléments de mathématique. Fasc. XXVI. Groupes et algèbres de Lie. Chapitre I: Algèbres de Lie, Seconde édition. Actualités Scientifiques et Industrielles, No. 1285, Hermann, Paris, 1971 (French). · Zbl 0213.04103 [4] Alfred S. Cavaretta, Wolfgang Dahmen, and Charles A. Micchelli, Stationary subdivision, Mem. Amer. Math. Soc. 93 (1991), no. 453, vi+186. · Zbl 0741.41009 [5] A. Cohen, Ondelettes, analyses multiresolutions et traitement numerique du signal, PhD thesis, Universite Paris IX, Dauphine, 1990. [6] Albert Cohen and Ingrid Daubechies, A stability criterion for biorthogonal wavelet bases and their related subband coding scheme, Duke Math. J. 68 (1992), no. 2, 313 – 335. · Zbl 0784.42022 [7] A. Cohen, Ingrid Daubechies, and J.-C. Feauveau, Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math. 45 (1992), no. 5, 485 – 560. · Zbl 0776.42020 [8] Lawrence J. Corwin and Frederick P. Greenleaf, Representations of nilpotent Lie groups and their applications. Part I, Cambridge Studies in Advanced Mathematics, vol. 18, Cambridge University Press, Cambridge, 1990. Basic theory and examples. · Zbl 0704.22007 [9] Xingde Dai and David R. Larson, Wandering vectors for unitary systems and orthogonal wavelets, Mem. Amer. Math. Soc. 134 (1998), no. 640, viii+68. · Zbl 0990.42022 [10] Ingrid Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), no. 7, 909 – 996. · Zbl 0644.42026 [11] 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. · Zbl 0776.42018 [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. · Zbl 0645.42010 [13] J. Dixmier and W. G. Lister, Derivations of nilpotent Lie algebras, Proceedings of the American Mathematical Society, 8 (1957), 155-158. · Zbl 0079.04802 [14] Joan L. Dyer, A nilpotent Lie algebra with nilpotent automorphism group, Bull. Amer. Math. Soc. 76 (1970), 52 – 56. · Zbl 0198.05402 [15] Nira Dyn, John A. Gregory, and David Levin, Analysis of uniform binary subdivision schemes for curve design, Constr. Approx. 7 (1991), no. 2, 127 – 147. · Zbl 0724.41011 [16] Timo Eirola, Sobolev characterization of solutions of dilation equations, SIAM J. Math. Anal. 23 (1992), no. 4, 1015 – 1030. · Zbl 0761.42014 [17] G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), no. 2, 161 – 207. · Zbl 0312.35026 [18] G. B. Folland, Harmonic Analysis in Phase Space, Princeton University Press, Princeton, New Jersey, 1989. · Zbl 0682.43001 [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). · Zbl 0064.35501 [21] A. Haar, Zur Theorie der orthogonalen Funktionensysteme, Mathematische Annalen, 69 (1910), 331-371. · JFM 41.0469.03 [22] SigurÄ’ur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. [23] Peter Niels Heller and Raymond O. Wells Jr., The spectral theory of multiresolution operators and applications, Wavelets: theory, algorithms, and applications (Taormina, 1993) Wavelet Anal. Appl., vol. 5, Academic Press, San Diego, CA, 1994, pp. 13 – 31. · Zbl 0845.42018 [24] L. Hervé, Construction et regularite des fonctions d’ echelle, SIAM J. Math. Anal., 5 (1995), 26. [25] Robert Hermann, Lie groups for physicists, W. A. Benjamin, Inc., New York-Amsterdam, 1966. · Zbl 0135.06901 [26] James E. Humphreys, Introduction to Lie algebras and representation theory, Springer-Verlag, New York-Berlin, 1972. Graduate Texts in Mathematics, Vol. 9. · Zbl 0254.17004 [27] N. Jacobson, A note on automorphisms and derivations of Lie algebras, Proceedings of the American Mathematical Society, 6 (1995), 281-283. · Zbl 0064.27002 [28] Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons), New York-London, 1962. · Zbl 0121.27504 [29] Rong Qing Jia, Subdivision schemes in \?_{\?} spaces, Adv. Comput. Math. 3 (1995), no. 4, 309 – 341. · Zbl 0833.65148 [30] Rong Qing Jia and Zuowei Shen, Multiresolution and wavelets, Proc. Edinburgh Math. Soc. (2) 37 (1994), no. 2, 271 – 300. · Zbl 0809.42018 [31] S. Kakutani, Über die Metrization der Topologischen Gruppen, Proceedings of the Imperial Academay of Tokyo, 12 (1936), 82-84. · JFM 62.1230.03 [32] Wayne M. Lawton, Tight frames of compactly supported affine wavelets, J. Math. Phys. 31 (1990), no. 8, 1898 – 1901. · Zbl 0708.46020 [33] Wayne M. Lawton, Necessary and sufficient conditions for constructing orthonormal wavelet bases, J. Math. Phys. 32 (1991), no. 1, 57 – 61. · Zbl 0757.46012 [34] Wayne M. Lawton, Multiresolution properties of the wavelet Galerkin operator, J. Math. Phys. 32 (1991), no. 6, 1440 – 1443. · Zbl 0733.46004 [35] W. Lawton and H. Resnikoff, Multidimensional Wavelet Bases, Technical Report, AWARE, Inc., Bedford, Massachusettes, 1991. [36] W. Lawton, S. L. Lee, and Zuowei Shen, Stability and orthonormality of multivariate refinable functions, SIAM J. Math. Anal. 28 (1997), no. 4, 999 – 1014. · Zbl 0872.41003 [37] W. Lawton, S. L. Lee, and Zuowei Shen, Convergence of multidimensional cascade algorithm, Numer. Math. 78 (1998), no. 3, 427 – 438. · Zbl 0890.41005 [38] Pierre Gilles Lemarié, Base d’ondelettes sur les groupes de Lie stratifiés, Bull. Soc. Math. France 117 (1989), no. 2, 211 – 232 (French, with English summary). · Zbl 0711.43004 [39] Rui Lin Long and Di Rong Chen, Biorthogonal wavelet bases on \?^{\?}, Appl. Comput. Harmon. Anal. 2 (1995), no. 3, 230 – 242. · Zbl 0846.42018 [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. [41] Krzysztof Maurin, General eigenfunction expansions and unitary representations of topological groups, Monografie Matematyczne, Tom 48, PWN-Polish Scientific Publishers, Warsaw, 1968. · Zbl 0185.39001 [42] Yves Meyer, Wavelets and operators, Cambridge Studies in Advanced Mathematics, vol. 37, Cambridge University Press, Cambridge, 1992. Translated from the 1990 French original by D. H. Salinger. · Zbl 0776.42019 [43] Peter J. Olver, Applications of Lie groups to differential equations, 2nd ed., Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1993. · Zbl 0785.58003 [44] Walter Rudin, Functional analysis, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. · Zbl 0253.46001 [45] Walter Rudin, Real and complex analysis, 2nd ed., McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. McGraw-Hill Series in Higher Mathematics. · Zbl 0278.26001 [46] Hans Samelson, Notes on Lie algebras, 2nd ed., Universitext, Springer-Verlag, New York, 1990. · Zbl 0708.17005 [47] O. L. Sattinger and O. L. Weaver, Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics, Springer-Verlag, New York, 1986. · Zbl 0595.22017 [48] Zuowei Shen, Refinable function vectors, SIAM J. Math. Anal. 29 (1998), no. 1, 235 – 250. · Zbl 0913.42028 [49] I. J. Schoenberg, Contributions to the problem of approximation of equidistant data by analytic functions, Quarterly Applied Mathematics., 4 (1946), 112-141. [50] L. Schwartz, Theorie des distributions, Hermann, Paris, 1957. [51] George Fix and Gilbert Strang, Fourier analysis of the finite element method in Ritz-Galerkin theory., Studies in Appl. Math. 48 (1969), 265 – 273. · Zbl 0179.22501 [52] Robert S. Strichartz, A guide to distribution theory and Fourier transforms, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1994. · Zbl 0854.46035 [53] Robert S. Strichartz, Self-similarity on nilpotent Lie groups, Geometric analysis (Philadelphia, PA, 1991) Contemp. Math., vol. 140, Amer. Math. Soc., Providence, RI, 1992, pp. 123 – 157. · Zbl 0797.43004 [54] Michael E. Taylor, Noncommutative harmonic analysis, Mathematical Surveys and Monographs, vol. 22, American Mathematical Society, Providence, RI, 1986. · Zbl 0604.43001 [55] François Trèves, Topological vector spaces, distributions and kernels, Academic Press, New York-London, 1967. · Zbl 0171.10402 [56] V. S. Varadarajan, Lie groups, Lie algebras, and their representations, Graduate Texts in Mathematics, vol. 102, Springer-Verlag, New York, 1984. Reprint of the 1974 edition. · Zbl 0955.22500 [57] Lars F. Villemoes, Energy moments in time and frequency for two-scale difference equation solutions and wavelets, SIAM J. Math. Anal. 23 (1992), no. 6, 1519 – 1543. · Zbl 0759.39005 [58] K. Yoshida, Functional Analysis, Springer, New York, 1980.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.