The Path-Connectivity of s-Elementary Tight Frame Wavelets Xingde Dai and Yuanan Diao Abstract. An s-elementary tight frame wavelet is a tight frame wavelet whose Fourier transform is of the form 1 √ 2π χ E for some measurable set E ⊂ R. It is known that the frame bound of an s-elementary tight frame wavelet is an integer. In this paper, we prove that for any integer k ≥ 1, the set of all s-elementary tight frame wavelets of frame bound k is a path-connected component of the set of all s-elementary tight frame wavelets (which is itself a non path-connected set). 1. Introduction The topological property of various families of wavelets is an inter- esting topic in the study of wavelet theory. The question concerning the path-connectedness of the set of all orthonormal wavelets was first raised in [7]. Similar questions were raised and studied in [5, 8, 9, 10, 11] about the set of all MRA-wavelets, tight frame wavelets, MRA tight frame wavelets and s-elementary frame wavelets. In [8, 11], it is shown that the set of MRA-wavelets is path-connected. In [10], it is shown that the set of s-elementary wavelets are path-connected. The proofs of these theorems were based on the complete character- izations of the MRA-wavelets and s-elementary wavelets. While the complete characterization of the s-elementary frame wavelets is still an open question, it has been shown that the set of s-elementary frame wavelets is path-connected as well [5]. In this paper, we will prove the path-connectedness of the s-elementary tight frame wavelets (with the same frame bounds). A set S ⊂ L 2 (R) is said to be path-connected under the norm topology of L 2 (R) if for any two members f , g ∈ S , there exists a Yuanan Diao is partially supported by NSF grant DMS-0310562. 1