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The Fourier coefficients of the spectral measures

For tex2html_wrap_inline670 , the functional tex2html_wrap_inline672 defines by Riesz representation theorem a measure tex2html_wrap_inline584 on [-1,1], which is the spectral measure of tex2html_wrap_inline678 . On the circle tex2html_wrap_inline680 , we are interested in the spectral measures tex2html_wrap_inline682 with respect to the unitary operators tex2html_wrap_inline628 which are also determined by their Fourier coefficients tex2html_wrap_inline686 . Let tex2html_wrap_inline688 be the spectral measure of tex2html_wrap_inline678 with respect to tex2html_wrap_inline576 . These measures are related as follows:

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Remarks.
1) The measure tex2html_wrap_inline534 on tex2html_wrap_inline680 is a spectral measure of tex2html_wrap_inline716 of the unitary operator tex2html_wrap_inline718 on tex2html_wrap_inline574 which gives a the simultaneous evolution of tex2html_wrap_inline722 and tex2html_wrap_inline724 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 tex2html_wrap_inline730 by other moments tex2html_wrap_inline732 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]).

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Oliver Knill
Tue Aug 18, 1998