Characterization of Smoothness of Multivariate Refinable Functions in Sobolev Spaces

Jia, Rong-Qing

doi:10.1090/S0002-9947-99-02185-6

Characterization of Smoothness of Multivariate Refinable Functions in Sobolev Spaces
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by Rong-Qing Jia

Trans. Amer. Math. Soc. 351 (1999), 4089-4112

DOI: https://doi.org/10.1090/S0002-9947-99-02185-6

Published electronically: July 1, 1999

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Abstract:

Wavelets are generated from refinable functions by using mul tiresolution analysis. In this paper we investigate the smoothness properties of multivariate refinable functions in Sobolev spaces. We characterize the optimal smoothness of a multivariate refinable function in terms of the spectral radius of the corresponding transition operator restricted to a suitable finite dimensional invariant subspace. Several examples are provided to illustrate the general theory.

References

C. de Boor, K. Höllig, and S. Riemenschneider, Box splines, Applied Mathematical Sciences, vol. 98, Springer-Verlag, New York, 1993. MR 1243635, DOI 10.1007/978-1-4757-2244-4
Alfred S. Cavaretta, Wolfgang Dahmen, and Charles A. Micchelli, Stationary subdivision, Mem. Amer. Math. Soc. 93 (1991), no. 453, vi+186. MR 1079033, DOI 10.1090/memo/0453
Albert Cohen and Ingrid Daubechies, Nonseparable bidimensional wavelet bases, Rev. Mat. Iberoamericana 9 (1993), no. 1, 51–137. MR 1216125, DOI 10.4171/RMI/133
Albert Cohen and Ingrid Daubechies, A new technique to estimate the regularity of refinable functions, Rev. Mat. Iberoamericana 12 (1996), no. 2, 527–591. MR 1402677, DOI 10.4171/RMI/207
Stephan Dahlke, Wolfgang Dahmen, and Vera Latour, Smooth refinable functions and wavelets obtained by convolution products, Appl. Comput. Harmon. Anal. 2 (1995), no. 1, 68–84. MR 1313100, DOI 10.1006/acha.1995.1006
Ronald A. DeVore and George G. Lorentz, Constructive approximation, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 303, Springer-Verlag, Berlin, 1993. MR 1261635
Timo Eirola, Sobolev characterization of solutions of dilation equations, SIAM J. Math. Anal. 23 (1992), no. 4, 1015–1030. MR 1166573, DOI 10.1137/0523058
T. N. T. Goodman, Charles A. Micchelli, and J. D. Ward, Spectral radius formulas for subdivision operators, Recent advances in wavelet analysis, Wavelet Anal. Appl., vol. 3, Academic Press, Boston, MA, 1994, pp. 335–360. MR 1244611
K. Gröchenig and W. R. Madych, Multiresolution analysis, Haar bases, and self-similar tilings of $\textbf {R}^n$, IEEE Trans. Inform. Theory 38 (1992), no. 2, 556–568. MR 1162214, DOI 10.1109/18.119723
B. Han and R. Q. Jia, Multivariate refinement equations and subdivision schemes, SIAM J. Math. Anal. 29 (1998), 1177–1999.
Tom Hogan, Stability and independence of the shifts of a multivariate refinable function, Approximation theory VIII, Vol. 2 (College Station, TX, 1995) Ser. Approx. Decompos., vol. 6, World Sci. Publ., River Edge, NJ, 1995, pp. 159–166. MR 1471783
Rong Qing Jia, The Toeplitz theorem and its applications to approximation theory and linear PDEs, Trans. Amer. Math. Soc. 347 (1995), no. 7, 2585–2594. MR 1277117, DOI 10.1090/S0002-9947-1995-1277117-8
Rong Qing Jia, Subdivision schemes in $L_p$ spaces, Adv. Comput. Math. 3 (1995), no. 4, 309–341. MR 1339166, DOI 10.1007/BF03028366
R. Q. Jia, Refinable shift-invariant spaces: from splines to wavelets, in Approximation Theory VIII, Vol. 2, C. K. Chui and L. L. Schumaker (eds.), World Scientific Publishing Co., Inc., 1995, pp. 179–208.
R. Q. Jia, The subdivision and transition operators associated with a refinement equation, in Advanced Topics in Multivariate Approximation, F. Fontanella, K. Jetter and P.-J. Laurent (eds.), World Scientific Publishing Co., Inc., 1996, pp. 139–154.
Rong-Qing Jia, Approximation properties of multivariate wavelets, Math. Comp. 67 (1998), no. 222, 647–665. MR 1451324, DOI 10.1090/S0025-5718-98-00925-9
Rong Qing Jia and Charles A. Micchelli, On linear independence for integer translates of a finite number of functions, Proc. Edinburgh Math. Soc. (2) 36 (1993), no. 1, 69–85. MR 1200188, DOI 10.1017/S0013091500005903
R.-Q. Jia and C. A. Micchelli, Using the refinement equation for the construction of pre-wavelets. V. Extensibility of trigonometric polynomials, Computing 48 (1992), no. 1, 61–72 (English, with German summary). MR 1162384, DOI 10.1007/BF02241706
Rong Qing Jia and Jianzhong Wang, Stability and linear independence associated with wavelet decompositions, Proc. Amer. Math. Soc. 117 (1993), no. 4, 1115–1124. MR 1120507, DOI 10.1090/S0002-9939-1993-1120507-8
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. MR 1453317, DOI 10.1137/S003614109528815X
S. D. Riemenschneider and Z. W. Shen, Multidimensional interpolatory subdivision schemes, SIAM J. Numer. Math. 34 (1997), 2357–2381.
Amos Ron, Smooth refinable functions provide good approximation orders, SIAM J. Math. Anal. 28 (1997), no. 3, 731–748. MR 1443617, DOI 10.1137/S0036141095282486
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Lars F. Villemoes, Sobolev regularity of wavelets and stability of iterated filter banks, Progress in wavelet analysis and applications (Toulouse, 1992) Frontières, Gif-sur-Yvette, 1993, pp. 243–251. MR 1282938
Lars F. Villemoes, Wavelet analysis of refinement equations, SIAM J. Math. Anal. 25 (1994), no. 5, 1433–1460. MR 1289147, DOI 10.1137/S0036141092228179
Lars F. Villemoes, Continuity of nonseparable quincunx wavelets, Appl. Comput. Harmon. Anal. 1 (1994), no. 2, 180–187. MR 1310642, DOI 10.1006/acha.1994.1005
Ding-Xuan Zhou, Stability of refinable functions, multiresolution analysis, and Haar bases, SIAM J. Math. Anal. 27 (1996), no. 3, 891–904. MR 1382838, DOI 10.1137/0527047