Baggett, Lawrence W.; Larsen, Nadia S.; Merrill, Kathy D.; Packer, Judith A.; and Raeburn, Iain F., 2009, Generalized multiresolution analyses with given multiplicity functions, Journal of Fourier Analysis and Applications, 15(5), 616-633.
Generalized multiresolution analyses are increasing sequences of subspaces of a Hilbert space H that fail to be multiresolution analyses in the sense of wavelet theory because the core subspace does not have an orthonormal basis generated by a fixed scaling function. Previous authors have studied a multiplicity function m which, loosely speaking, measures the failure of the GMRA to be an MRA. When the Hilbert space H is L2(Rn), the possible multiplicity functions have been characterized by Baggett and Merrill. Here we start with a function m satisfying a consistency condition, which is known to be necessary, and build a GMRA in an abstract Hilbert space with multiplicity function m.