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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.

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