The s-elementary frame wavelets are path connected

Abstract

An s-elementary frame wavelet is a function ψ ∈ L 2 ( R ) \psi \in L^2(\mathbb {R}) which is a frame wavelet and is defined by a Lebesgue measurable set E ⊂ R E\subset \mathbb {R} such that ψ ^ = 1 2 π χ E \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 ( R ) L^2(\mathbb {R}) -norm. This result also holds for s-elementary A A -dilation frame wavelets in L 2 ( R d ) 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.