Convergence of cascade sequence on the Heisenberg group

Liu, Heping; Liu, Yu

doi:10.1090/S0002-9939-05-08108-6

Convergence of cascade sequence on the Heisenberg group
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by Heping Liu and Yu Liu PDF

Proc. Amer. Math. Soc. 134 (2006), 1413-1423 Request permission

Abstract:

The investigation of convergence of cascade sequence plays an important role in wavelet analysis on the Euclidean space and also in wavelet analysis on the Heisenberg group. This paper characterizes the $L^p(\mathbf H^d)$ $(1\leq p\leq \infty )$-convergence of cascade sequence on the Heisenberg group in terms of the $p$-norm joint spectral radius of a collection of matrices associated with the refinement sequence and gives a sufficient condition.

References

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. MR 1317525, DOI 10.2977/prims/1195164793
Bin Han and Rong-Qing Jia, Multivariate refinement equations and convergence of subdivision schemes, SIAM J. Math. Anal. 29 (1998), no. 5, 1177–1199. MR 1618691, DOI 10.1137/S0036141097294032
Rong Qing Jia, Subdivision schemes in $L_p$ spaces, Adv. Comput. Math. 3 (1995), no. 4, 309–341. MR 1339166, DOI 10.1007/BF03028366
W. Lawton, S. L. Lee, and Zuowei Shen, Convergence of multidimensional cascade algorithm, Numer. Math. 78 (1998), no. 3, 427–438. MR 1603354, DOI 10.1007/s002110050319
Wayne Lawton, Infinite convolution products and refinable distributions on Lie groups, Trans. Amer. Math. Soc. 352 (2000), no. 6, 2913–2936. MR 1638258, DOI 10.1090/S0002-9947-00-02409-0
Heping Liu and Lizhong Peng, Admissible wavelets associated with the Heisenberg group, Pacific J. Math. 180 (1997), no. 1, 101–123. MR 1474896, DOI 10.2140/pjm.1997.180.101
Lizhong Peng, Wavelets on the Heisenberg group, Geometry and nonlinear partial differential equations (Hangzhou, 2001) AMS/IP Stud. Adv. Math., vol. 29, Amer. Math. Soc., Providence, RI, 2002, pp. 123–131. MR 1926440, DOI 10.1090/amsip/029/12
Gian-Carlo Rota and Gilbert Strang, A note on the joint spectral radius, Nederl. Akad. Wetensch. Proc. Ser. A 63 = Indag. Math. 22 (1960), 379–381. MR 0147922, DOI 10.1016/S1385-7258(60)50046-1
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. MR 1197594, DOI 10.1090/conm/140/1197594
Q. D. Yang, Multiresolution analysis on non-abelian locally compact groups, Ph.D. Thesis, Dept. of Math., Univ. of Saskatchewan, 1999.