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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]).
