Unitary extension principle on~zero-dimensional~locally~compact~groups

S. F. Lukomskii ; Iu. S. Kruss

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 23 (2023) no. 3, pp. 320-338

Résumé

In this article, we obtain methods for constructing step tight frames on an arbitrary locally compact zero-dimensional group. To do this, we use the principle of unitary extension. First, we indicate a method for constructing a step scaling function on an arbitrary zero-dimensional group. To construct the scaling function, we use an oriented tree and specify the conditions on the tree under which the tree generates the mask $m_0$ of a scaling function. Then we find conditions on the masks $m_0, m_1,\ldots , m_q$ under which the corresponding wavelet functions $\psi_1, \psi_2,\ldots ,\psi_q$ generate a tight frame. Using these conditions, we indicate a way of constructing such masks. In conclusion, we give examples of the construction of tight frames.