Parseval frame wavelet multipliers in L 2(ℝ d )

Abstract

Let A be a d × d real expansive matrix. An A-dilation Parseval frame wavelet is a function ψ ∈ L 2(ℝ d ), such that the set $$ \left\{ {\left| {\det A} \right|^{\frac{n} {2}} \psi \left( {A^n t - \ell } \right):n \in \mathbb{Z},\ell \in \mathbb{Z}^d } \right\} $$ forms a Parseval frame for L 2(ℝ d ). A measurable function f is called an A-dilation Parseval frame wavelet multiplier if the inverse Fourier transform of dψ⌃ is an A-dilation Parseval frame wavelet whenever ψ is an A-dilation Parseval frame wavelet, where ψ⌃ denotes the Fourier transform of ψ. In this paper, the authors completely characterize all A-dilation Parseval frame wavelet multipliers for any integral expansive matrix A with |det(A)| = 2. As an application, the path-connectivity of the set of all A-dilation Parseval frame wavelets with a frame MRA in L 2(ℝ d ) is discussed.