Multiwavelet frames from refinable function vectors. (English) Zbl 1059.42030

A \(d\)-refinable function vector \(\phi = (\phi_1, \ldots, \phi_r)^T\) satisfies the refinement equation \[ \phi = | d| \sum_{k \in {\mathbb Z}} a_k \phi(d\cdot - k), \] where \(r\) is the multiplicity, \(d\) the dilation factor, and \(a\) is the finitely supported sequence of \(r\times r\) matrices. Such a vector generates a wavelet function vector by \[ \widehat \psi^l (d \xi) = b^l(\xi) \widehat \phi(\xi), \] where \(l=1, \ldots, L\) and \(b^l\) are periodic trigonometric polynomials.
The paper under review studies the properties of dual wavelet frames and of wavelet frames generated by refinable function vectors. In particular, algorithms for constructing dual wavelet frames with maximum vanishing moments are given. Moreover, symmetry/antisymmetry properties of wavelets are examined and examples are provided.


42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42C15 General harmonic expansions, frames
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