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Stationary vector subdivision: quotient ideals, differences and approximation power. (English) Zbl 1057.42029

The stationary vector subdivision operator \( S_A = S_{M,A} \) based on an expanding matrix \( M\in \mathbb Z^{s\times s} \) and a finitely supported matrix valued mask \( A \in l_p^{N \times N}(\mathbb Z^s) \) is defined for any \( c \in l^N_p (\mathbb Z )\) by \[ S_{M,A} c := A \star_M c := \sum_{\beta \in \mathbb Z^s}A(\cdot - M \beta) c(\beta) . \] In this paper, those subdivision schemes which possess polynomial eigensequences are characterized in terms of a difference operator. Moreover, convergent stationary subdivision schemes are connected to a factorization property which is expressed in terms of quotient ideals. These results can be applied in order to describe whether a subdivision operator \( S_A\) preserves polynomial sequences of a certain total degree.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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