Spectral distribution of generalized Kac–Murdock–Szegö matrices

If ζ is a nonzero complex number and P is a monic polynomial with real coefficients, let Kn(ζ;P)=(P(|r−s|)ρ|r−s|ei(r−s)φ)r,s=1n. We call the class of matrices Tn=∑jcjKn(ζj;Pj) (cj real, finite sum) generalized Kac–Murdock–Szego matrices. If |ζj|<1 for all j, the family {Tn} has a generating function in C[−π,π], and Szego's distribution theorem implies that the eigenvalues of Tn are distributed like the values of g as n→∞. However, Szego's theorem does not apply if |ζj|⩾1 for some j. Nevertheless, we show that in this case, provided that Pj is even if |ζj|=1, there is a function g∈C[−π,π] such that all but a finite number (independent of n) of the eigenvalues of Tn are distributed like the values of g as n→∞. We also discuss the asymptotic behavior of the remaining eigenvalues as n→∞; however, a complete resolution of this question is not yet available.