Unitary Correlations and the Fejér Kernel

Let M be a unitary matrix with eigenvalues tj, and let f be a function on the unit circle. Define Xf(M)=∑f(tj). We derive exact and asymptotic formulae for the covariance of Xf and Xg with respect to the measures |χ(M)|2 dM where dM is Haar measure and χ an irreducible character. The asymptotic results include an analysis of the Fejér kernel which may be of independent interest.

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