Mean-square approximation by polynomials on the unit disk

Thomas L. Kriete, Barbara D. MacCluer
1990 Transactions of the American Mathematical Society  
We investigate p2(fJ) , the closure in L2(fJ) of the complex polynomials, for certain measures fJ on the closed unit disk in the complex plane. Specifically, we present a condition on fJ which guarantees that p2(fJ) decomposes into a natural direct sum. 1 il-J log log G(r) dr (some small J). Let us always use the term arc to mean a non empty open arc in {) D . Theorem D. Suppose the integral (1.2) converges (and G satisfies a mild regularity condition). If log W ELI (I) for some arc I c {)D,
more » ... n P2(fl) does not split. This theorem, which is proved in §7 below, improves results from [KrI] and [Trl]. The best result in the converse direction (indeed, it is best possible) is due to A. L. Vol' berg and is based on his work on quasi analyticity [V3, VEl. Theorem 1.1 (Vol' berg). Suppose that the integral (1.2) diverges (and G satisfies mild regularity conditions). If logw tt-LI , then P2(fl) splits. This result can be derived from Theorem 1 in [VEl in the same way that Theorem 5 and Corollary 2 in [Kr1] are deduced from the Levinson-Beurling Theorem. In this paper, then, we will be concerned only with the situation in which G decreases to zero slowly enough that the integral (1.2) is finite, or even with the case where G does not decrease at all, e.g., G(r) == 1 . Note that this includes the standard Bergman space weights G(r) = (1r)" , a > -1 , weights which decay exponentially like ( 1.3) G(r) = exp (-(h) Q) , a> 0, License or copyright restrictions may apply to redistribution; see I Q( <5 , e) log -> 2NI . J J e J Setting e = e) we have, for each k = 1,2, ... , n (using Lemma 6.1),
doi:10.1090/s0002-9947-1990-0948193-x fatcat:4smuft7kizf7nf4udp6snmz224