Steklov Institute of Mathematics at St.Petersburg

PREPRINT 23/2004

Vladimir Kapustin and Alexei Poltoratski


This preprint was accepted December 26, 2004

In [1], D.~Clark considered certain families of positive singular 
measures on the unit circle associated to inner functions $\theta$  
in the unit disk. These measures were shown to be the spectral measures 
of unitary rank-one perturbations of the model operator acting on the 
(scalar) model space $K_\theta$, a subspace of the  Hardy space $H^2$. 
In particular, it was shown that  $K_\theta$ can be mapped unitarily 
onto $L^2(\sigma)$ for any of such measures $\sigma$. Later  it was 
proved [2] that this mapping, which could be viewed as a generalization 
of the classical Fourier transform, actually takes functions from 
$K_\theta$ to their angular boundary values, which exist $\sigma$@-almost 
everywhere for any such $\sigma$. In this paper we present an analog of 
the above-mentioned results for vector-valued model spaces.
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