## Illinois Journal of Mathematics

### Extremal problems in Bergman spaces and an extension of Ryabykh’s theorem

Timothy Ferguson

#### Abstract

We study linear extremal problems in the Bergman space $A^p$ of the unit disc for $p$ an even integer. Given a functional on the dual space of $A^p$ with representing kernel $k \in A^q$, where $1/p + 1/q = 1$, we show that if the Taylor coefficients of $k$ are sufficiently small, then the extremal function $F \in H^{\infty}$. We also show that if $q \le q_1 < \infty$, then $F \in H^{(p-1)q_1}$ if and only if $k \in H^{q_1}$.

#### Article information

Source
Illinois J. Math., Volume 55, Number 2 (2011), 555-573.

Dates
First available in Project Euclid: 1 February 2013

https://projecteuclid.org/euclid.ijm/1359762402

Digital Object Identifier
doi:10.1215/ijm/1359762402

Mathematical Reviews number (MathSciNet)
MR3020696

Zentralblatt MATH identifier
1276.30062

Subjects
Primary: 30D60: Quasi-analytic and other classes of functions 30D65

#### Citation

Ferguson, Timothy. Extremal problems in Bergman spaces and an extension of Ryabykh’s theorem. Illinois J. Math. 55 (2011), no. 2, 555--573. doi:10.1215/ijm/1359762402. https://projecteuclid.org/euclid.ijm/1359762402

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