Wavelets and the Angle between Past and Future

The main result, the Riesz projectionP+(or, equivalently, Hilbert TransformT), is bounded in the weighted spaceL2(W) whereWis a matrix-valued weight if and only ifsupi‖[1|I|∫IW]1/2[1|I|∫IW−1]1/2‖<∞wheresupremumis taken over all intervalsI. Motivation for this problem comes from stationary processes (Riesz projection is bounded means the angle between “past” and “future” of a stationary process with spectral measureWis nonzero). In the scalar case the result is the well known Hunt–Muckenhoupt–Wheeden theorem. The main step in our proof is to show that a vector Haar system forms an unconditional basis inL2(W). As a byproduct of our approach we get some new results about bases of wavelets in weighted spaces (in both scalar and vector-valued cases).