关键词: Locally compact group, unitary representation, continuous frame, wavelet transform, direct integral

Abstract:

Let π be a unitary representation of a locally compact topological group G on a separable Hilbert space H. A vector Ψ ∈ H is called a continuous frame wavelet if there exist A, B > 0 such that
A||Φ||² ≤∫_G|<π(g)Ψ ,Φ|²dg ≤ B||Φ||²   (Φ ∈ H),

in which dg is the left Haar measure of G. Similar to the study of wavelets, an essential problem in the study of continuous frame wavelets is how to characterize them under the given unitary representation. Moreover, we investigate a relation between admissible vectors of π and its components.

Key words: Locally compact group, unitary representation, continuous frame, wavelet transform, direct integral