Baggett’s Problem for Frame Wavelets

Abstract

Baggett’s problem asks whether every Parseval wavelet ψ is associated with a generalized multiresolution analysis (GMRA). Equivalently, one can ask whether the intersection of all dilates of the space V(ψ) of negative dilates of ψ must be necessarily trivial. We present the current state of knowledge and ramifications of this open problem for the wavelet theory. We also construct an example of (nontight) frame wavelet ψ with many desirable properties (such as smoothness, good decay, having a dual frame wavelet) such that its corresponding space of negative dilates is equal to the entire space V(ψ) = L²(ℝ). This improves the original example of this kind by Rzeszotnik and the author [Math. Ann. 332 (2005), 705–720].