lp-Norms of Fourier Coefficients of Powers of a Blaschke Factor

We determine the asymptotic behavior of the lp-norms of the sequence of Taylor coefficients of bn, where b(z) = z -. 1 - <overline>.z, |.| < 1, is an automorphism of the unit disk, p. [1,8], and n is large. It is known that in the parameter range p. [1, 2] a sharp upper bound bn lAp = cpn 2-p 2p holds. In this article we find that this estimate is valid even when p. [1, 4). We prove that bn lA4 = c4 log n n 14 and for p. (4,8] that bn lAp = cpn 1-p 3p. The upper bounds are shown to be asymptotically sharp as n tends to 8. As a direct application we prove the sharpness of existing upper estimates on analytic capacities in Beurling-Sobolev spaces. Our investigation is also motivated by a question of J. J. Sch <spacing diaeresis>affer about optimal estimates for norms of inverse matrices.