The Orthonormal Dilation Property for Abstract Parseval Wavelet Frames

Source abstract

Abstract. In this work we introduce a class of discrete groups containing subgroups of abstract translations and dilations, respectively. A variety of wavelet systems can appear as π(Γ)ψ , where π is a unitary representation of a wavelet group and Γ is the abstract pseudo-lattice Γ. We prove a sufficent condition in order that a Parseval frame π(Γ) ψ can be dilated to an orthonormal basis of the form τ (Γ) ψ, where τ is a super-representation of π. For a subclass of groups that includes the case where the translation subgroup is Heisenberg, we show that this condition always holds, and we cite familiar examples as applications.