Matrix Fourier multipliers for Parseval multi-wavelet frames

A Fourier multiplier for orthonormal wavelets is an L∞-function that sends every orthonormal wavelet to an orthonormal wavelet. This type of multipliers plays an important role in the study of basic properties of wavelets including some geometrical and topological properties of the wavelet theory. Matrix Fourier multipliers are matrices with L∞-function entries that map Parseval multi-wavelet frames to Parseval multi-wavelet frames. Like Fourier wavelet multiplier, matrix Fourier multipliers can be used to derive new multi-wavelet frames and can help us better understand the basic theory of multi-wavelet frame theory. In this paper we characterize all the matrix Fourier multipliers for Parseval multi-wavelet frames.