Abstract
We study multivariate trigonometric polynomials satisfying the “sum-rule” conditions of a certain order. Based on the polyphase representation of these polynomials relative to a general dilation matrix, we develop a simple constructive method for a special type of decomposition of such polynomials. These decompositions are of interest in the analysis of convergence and smoothness of multivariate subdivision schemes associated with general dilation matrices. The approach presented in this paper leads directly to constructive algorithms, and is an alternative to the analysis of multivariate subdivision schemes in terms of the joint spectral radius of certain operators. Our convergence results apply to arbitrary dilation matrices, while the smoothness results are limited to two classes of dilation matrices.
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Cohen, A., Gröchenig, K., Villemoes, L.F.: Regularity of multivariate refinable functions. Constr. Approx. 15(2), 241–255 (1999)
Cavaretta, A.S., Dahmen, W., Micchelli, C.A.: Stationary subdivision. Mem. Am. Math. Soc. 93(453), vi+186 (1991)
Charina, M.: Vector multivariate subdivisio0n schemes: comparison of spectral methods for their regularity analysis. Appl. Comput. Harmon. Anal. (2011, in press)
Chen, D.-R., Jia, R.-Q., Riemenschneider, S.D.: Convergence of vector subdivision schemes in Sobolev spaces. Appl. Comput. Harmon. Anal. 12(1), 128–149 (2002)
Dyn, N.: Subdivision schemes in computer aided geometric design. In: Light, W. (ed.) Advances in Numerical Analysis, vol. II. Wavelets, Subdivision Algorithms and Radial Basis Functions, pp. 36–104. Clarendon Press, Oxford (1992)
Jia, R.Q.: Approximation properties of multivariate wavelets. Math. Comput. 67, 647–655 (1998)
Han, B.: Computing the smoothness exponent of a symmetric multivariate refinable function. SIAM J. Matrix Anal. Appl. 24, 693–714 (2003)
Han, B.: Compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix. J. Comput. Appl. Math. 155, 43–67 (2003)
Han, B.: Vector cascade algorithms and refinable function vector in Sobolev spaces. J. Approx. Theory 124, 44–88 (2003)
Han, B., Jia, R.-Q.: Multivariate refinement equations and convergence of subdivision schemes. SIAM J. Math. Anal. 29(5), 1177–1999 (1998)
Latour, V., Müller, J., Nickel, W.: Stationry subdivision for general scaling matrices. Math. Z. 227, 645–661 (1998)
Möller, H.M., Sauer, T.: Multivariate refinable functions of high approximation order via quotient ideals of Laurent polynomials. Adv. Comput. Math. 20(1–3), 205–228 (2004)
Novikov, I.Ya., Protasov, V.Yu., Skopina, M.A.: Wavelet theory. In: Translations Mathematical Monographs, vol. 239. AMS, Providence (2011)
Sauer, T.: How to generate smoother refinable functions from given one In: Haussmann, W., Jetter, K., Reimer, M., Stökler, J. (eds.) Modern Developments in Multivariate Approximation. International Series of Numerical Mathematics, vol. 145, pp. 279–294 (2003)
Sauer, T.: Differentiability of multivariate refinable functions and factorization. Adv. Comput. Math. 26(1–3), 211–235 (2006)
Skopina, M.: On construction of multivariate wavelets with vanishing moments. ACHA 20(3), 375–390 (2006)
Wojtaszczyk, P.: A mathematical introduction to wavelets. Lond. Math. Soc. Stud. Texts, vol. 37, xii+261 pp. Cambridge University Press, Cambridge (1997)
Zhou, D.-X.: Norms concerning subdivision sequences and their// applications in wavelets. Appl. Comput. Harmon. Anal. 11(3), 329–346 (2001)
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Communicated by R.Q. Jia.
This research was supported by The Hermann Minkowski Center for Geometry at Tel-Aviv University. The second author is also supported by grants 09-01-00162 and 12-01-00216 of RFBR.
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Dyn, N., Skopina, M. Decompositions of trigonometric polynomials with applications to multivariate subdivision schemes. Adv Comput Math 38, 321–349 (2013). https://doi.org/10.1007/s10444-011-9239-7
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DOI: https://doi.org/10.1007/s10444-011-9239-7
Keywords
- Subdivision operator
- Refinable function
- Dilation matrix
- Polyphase components of trigonometric polynomials