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Decompositions of trigonometric polynomials with applications to multivariate subdivision schemes

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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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Authors and Affiliations

  1. Tel-Aviv University, Tel-Aviv, Israel

    Nira Dyn

  2. St. Petersburg State University, St. Petersburg, Russia

    Maria Skopina

Authors
  1. Nira Dyn
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  2. Maria Skopina
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Correspondence to Maria Skopina.

Additional information

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

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