Abstract. We prove that the spectral norm of a finite Toeplitz matrix can be estimated from below through the Fejér mean of the generating function. This result has applications to the problem of finding the most probable values of $\|A_n x\|$ in case $A_n$ is a large finite Toeplitz matrix and $x$ is uniformly distributed on the unit sphere of ${\bf C}^n$.

AMS Subject Classification (1991): 47B35; 15A12, 42A05, 60H25

Received July 29, 2002, and in revised form December 12, 2002. (Registered under 2938/2009.)