Narrow Results

This is partly a survey article. We present a survey of results related with the characterization of scaling functions of multiresolution analyses. Given a linear invertible map A : ℝⁿ → ℝⁿ such that A(ℤⁿ) ⊂ ℤⁿ and all (complex) eigenvalues of A have absolute value greater than one we give a characterization of scaling functions of a frame multiresolution analysis on some closed subspaces of L²(ℝⁿ) denoted by . As a corollary we obtain that if the density number of the set G, |G|_n > 0 at the origin is less than one then no any function g ∈ L¹(ℝⁿ) can be a scaling function of an -FMRA associated with A. Putting together results obtained in [22], [12] one observes that for any measurable A*-invariant set G of positive measure always exists some -MRA associated with A.