Abstract
Motivated by the concept of directionally adapted subdivision for the definition of shearlet multiresolution, the paper considers a generalized class of multivariate stationary subdivision schemes, where in each iteration step a scheme and a dilation matrix can be chosen from a given finite set. The standard questions of convergence and refinability will be answered as well as the continuous dependence of the resulting limit functions from the selection process. In addition, the concept of a canonical factor for multivariate subdivision schemes is introduced, which follows in a straightforward fashion from algebraic properties of the scaling matrix and takes the role of a smoothing factor for symbols.
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References
Cavaretta, A.S., Dahmen, W., Micchelli, C.A.: Stationary Subdivision, Memoirs of the AMS, vol. 93 (453). Amer. Math. Soc., Providence (1991)
Cox, D., Little, J., O’Shea, D.: Ideals, Varieties and Algorithms. Undergraduate Texts in Mathematics, 2nd edn. Springer, Heidelberg (1996)
Dahmen, W., Micchelli, C.A.: Biorthogonal wavelet expansion. Constr. Approx. 13, 294–328 (1997)
Derado, J.: Multivariate refinable interpolating functions. Appl. Comp. Harmonic Anal. 7, 165–183 (1999)
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)
Jia, R.Q.: Subdivision schemes in l p spaces. Advances Comput. Math. 3, 309–341 (1995)
Kutyniok, G., Sauer, T.: From wavelets to shearlets and back again. In: Schumaker, L.L. (ed.) Approximation Theory XII, San Antonio, TX, pp. 201–209. Nashboro Press, Nashville (2007/2008)
Kutyniok, G., Sauer, T.: Adaptive directional subdivision schemes and shearlet multiresolution analysis. SIAM J. Math. Anal. 41, 1436–1471 (2009)
Marcus, M., Minc, H.: A Survey of Matrix Theory and Matrix Inequalities. Prindle, Weber & Schmidt (1969); paperback reprint, Dover Publications (1992)
Micchelli, C.A.: Interpolatory subdivision schemes and wavelets. J. Approx. Theory 86, 41–71 (1996)
Micchelli, C.A., Sauer, T.: Regularity of multiwavelets. Advances Comput. Math. 7(4), 455–545 (1997)
Micchelli, C.A., Sauer, T.: On vector subdivision. Math. Z. 229, 621–674 (1998)
Möller, H.M., Sauer, T.: Multivariate refinable functions of high approximation order via quotient ideals of Laurent polynomials. Advances Comput. Math. 20, 205–228 (2004)
Sauer, T.: Stationary vector subdivision – quotient ideals, differences and approximation power. Rev. R. Acad. Cien. Serie A. Mat. 96, 257–277 (2002)
Sauer, T.: Lagrange interpolation on subgrids of tensor product grids. Math. Comp. 73, 181–190 (2004)
Sauer, T.: Differentiability of multivariate refinable functions and factorization. Advances Comput. Math. 25, 211–235 (2006)
Sauer, T.: Multivariate refinable functions, difference and ideals – a simple tutorial. J. Comput. Appl. Math. 221, 447–459 (2008)
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Sauer, T. (2012). Multiple Subdivision Schemes. In: Boissonnat, JD., et al. Curves and Surfaces. Curves and Surfaces 2010. Lecture Notes in Computer Science, vol 6920. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27413-8_41
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DOI: https://doi.org/10.1007/978-3-642-27413-8_41
Publisher Name: Springer, Berlin, Heidelberg
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