Spectral distribution of generalized Kac–Murdock–Szegö matrices Original Research Article

If ζ is a nonzero complex number and P is a monic polynomial with real coefficients, let Kn(ζ;P)=(P(r−s)ρr−sei(r−s)φ)r,s=1n. We call the class of matrices Tn=∑jcjKn(ζj;Pj) (cj real, finite sum) generalized Kac–Murdock–Szeg matrices. If ζj<1 for all j, the family {Tn} has a generating function in C[−π,π], and Szegöʹs distribution theorem implies that the eigenvalues of Tn are distributed like the values of g as n→∞. However, Szegöʹs theorem does not apply if ζjgreater-or-equal, slanted1 for some j. Nevertheless, we show that in this case, provided that Pj is even if ζj=1, there is a function gset membership, variantC[−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.