TY - JOUR
T1 - Decompositions of trigonometric polynomials with applications to multivariate subdivision schemes
AU - Dyn, Nira
AU - Skopina, Maria
N1 - Funding Information:
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.
PY - 2013/2
Y1 - 2013/2
N2 - 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.
AB - 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.
KW - Dilation matrix
KW - Polyphase components of trigonometric polynomials
KW - Refinable function
KW - Subdivision operator
UR - http://www.scopus.com/inward/record.url?scp=84874011229&partnerID=8YFLogxK
U2 - 10.1007/s10444-011-9239-7
DO - 10.1007/s10444-011-9239-7
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AN - SCOPUS:84874011229
SN - 1019-7168
VL - 38
SP - 321
EP - 349
JO - Advances in Computational Mathematics
JF - Advances in Computational Mathematics
IS - 2
ER -