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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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