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Parseval frame scaling sets and MSF Parseval frame wavelets. (English) Zbl 1198.42048
Summary: We consider the Parseval frame (PF) scaling sets and the MSF Parseval frame wavelets (PFWs) in $$L^{2}(\mathbb{R}^d)$$ with dilations induced by expanding matrices $$A$$ with integer coefficients of arbitrary determinant such that $$|\det A|=2$$. We firstly characterize the PF scaling sets, and then provide a method of construction of PF scaling sets. We also prove that all PF scaling sets arise in that way. Finally, by studying the relation between the MSF PFWs and the PF scaling sets, we derive that each PF scaling set $$S$$ gives rise to a MSF PFW $$\psi$$, and furthermore each MSF PFW whose dimension function is essentially bounded by 1 arises from a PF scaling set and the corresponding PF MRA. Using our results, one can easily construct various PF scaling sets and MSF PFWs.
Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

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