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For , the functional defines by Riesz representation theorem a measure on [-1,1], which is the spectral measure of . On the circle , we are interested in the spectral measures with respect to the unitary operators which are also determined by their Fourier coefficients . Let be the spectral measure of with respect to . These measures are related as follows:

Remarks.

1) The measure on
is a spectral measure of of the
unitary operator on which gives a
the simultaneous evolution of and on two copies of
the Hilbert space *H*.

2) The study of orthogonal polynomials on [-1,1] by lifting them
onto the circle goes back to Szegö [29].
In [7], it was suggested to replace ordinary moments
by other moments
in order to get information on the
spectral measures of operators. However, the case of Chebychev polynomials
treated here has been left out in [7].
We should note that Chebychev polynomials are also useful in
similar contexts like polynomial expansions of the Green functions
(see [21]).

Tue Aug 18, 1998