Abstract
Let M be a unitary matrix with eigenvalues t j , and let f be a function on the unit circle. Define X f (M)=∑f(t j ). We derive exact and asymptotic formulae for the covariance of X f and X g 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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Bump, D., Diaconis, P. & Keller, J.B. Unitary Correlations and the Fejér Kernel. Mathematical Physics, Analysis and Geometry 5, 101–123 (2002). https://doi.org/10.1023/A:1016200519958
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DOI: https://doi.org/10.1023/A:1016200519958