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:
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 . In , 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 . We should note that Chebychev polynomials are also useful in similar contexts like polynomial expansions of the Green functions (see ).