Trigonometric bases for matrix weighted -spaces

We give a complete characterization of 2π-periodic matrix weights W for which the vector-valued trigonometric system forms a Schauder basis for the matrix weighted space L p ( T ; W ) . Then trigonometric quasi-greedy bases for L p ( T ; W ) are considered. Quasi-greedy bases are systems for which the simple thresholding approximation algorithm converges in norm. It is proved that such a trigonometric basis can be quasi-greedy only for p = 2 , and whenever the system forms a quasi-greedy basis, the basis must actually be a Riesz basis.