Abstract

We give an asymptotic upper bound as $ntoinfty$ for the entropy integral where $p_n$ is the $n$th degree orthonormal polynomial with respect to a weight $w(x)$ on $[-1,1]$ which belongs to the SzegH{o} class. We also study two functionals closely related to the entropy integral. First, their asymptotic behavior is completely described for weights $w$ in the Bernstein class. Then, as for the entropy, we obtain asymptotic upper bounds for these two functionals when $w(x)$ belongs to the SzegH{o} class. In each case, we give conditions for these upper bounds to be attained.