Symmetric framelets. (English) Zbl 1037.42038

Given a separable Hilbert space \(H\), a frame is a sequence \(\{ e_j: j \in {\mathbb{Z}} \}\) of elements of \(H\), such that there exist constants \(0<A\leq B <\infty\) such that for all \(f\in H\), \[ A \| f \| ^2 \leq \sum_{j\in \mathbb{Z}} | \langle f, e_j \rangle | ^2 \leq B \| f\| ^2 . \] A frame is tight if \(A=B\). It is known that the problem of constructing tight frames of wavelets (i.e., frames of the form \(\{ \psi_{j,k}^l (x) = 2^{j/2} \psi^l (2^j x - k): j,k \in \mathbb{Z}, l=1, \ldots, n \}\)) which are generated by a multiresolution analysis, can be reduced to finding functions \(m_k\), \(k=0, \ldots, n\), which satisfy the equation \[ M(\omega) M^*(\omega) = Id, \] where \(M(\omega)(1,k) = m_k(\omega)\) and \(M(\omega)(2,k) = m_k(\omega+\pi)\). The paper under review uses this approach to characterize tight wavelet frames associated with two symmetric or antisymmetric compactly supported refinable functions. All refinable masks of length up to 6 that satisfy this criterion are found.


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