The s-elementary frame wavelets are path connected

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.