The s-elementary frame wavelets are path connected

Abstract:

An s-elementary frame wavelet is a function $\psi \in L^2(\mathbb {R})$ which is a frame wavelet and is defined by a Lebesgue measurable set $E\subset \mathbb {R}$ such that $\hat {\psi }= \frac {1}{\sqrt {2\pi }}\chi _E$. In this paper we prove that the family of s-elementary frame wavelets is a path-connected set in the $L^2(\mathbb {R})$-norm. This result also holds for s-elementary $A$-dilation frame wavelets in $L^2(\mathbb {R}^d)$ in general. On the other hand, we prove that the path-connectedness of s-elementary frame wavelets cannot be strengthened to uniform path-connectedness. In fact, the sets of normalized tight frame wavelets and frame wavelets are not uniformly path-connected either.