## Wavelets and the angle between past and future.(English)Zbl 0876.42027

Summary: The main result, the Riesz projection $$P_+$$ (or, equivalently, Hilbert transform $$T$$), is bounded in the weighted space $$L^2(W)$$, where $$W$$ is a matrix-valued weight if and only if $\sup_I\Biggl|\Biggl[ {1\over|I|} \int_I W\Biggr]^{1/2}\Biggl[ {1\over|I|} \int_I W^{-1}\Biggr]^{-1/2}\Biggr|<\infty,$ where the supremum is taken over all intervals $$I$$. 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 measure $$W$$ is 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 in $$L^2(W)$$. 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).

### MSC:

 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 44A15 Special integral transforms (Legendre, Hilbert, etc.) 46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
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