On the Eigenvalues of Correlation Matrices

We obtain upper and lower bounds for the maximum eigenvalue of the matrix $C_{n=1}(F)=\|c_{p,q};\,p,q=0,1,\dots,n\|$, with elements of the form
$$ c_{p,q}=c_{p-q}=\int_0^{2\pi}\exp\{i(p-q)\lambda\}F(d\lambda), $$
where $F(d\lambda)$ is the measure on the segment $[0,2\pi];$ for the sum of the $K<n+1$ largest eigenvalues we give an estimate for the number of eigenvalues with fixed sum. We give examples of the measure $F(d\lambda)$ to illustrate the equations obtained.