##### Department of Mathematics,

University of California San Diego

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### Final Defense

## David Scheinker

#### UCSD

## Bounded Analytic Functions on the Polydisc

##### Abstract:

Function theory of several complex variables is much less well understood than function theory of functions of one variable. One approach to attempting to bridge this divide is to study an analytic function f on the polydisc as follows. Fix a $1$-dimensional algebraic variety $V$ in $C^n$ and let $F$ denote the restriction of $f$ to $V$. Since $V$ is $1$-dimensional, $F$ behaves somewhat like a function of one complex variable and we apply the theory of functions of one variable to $F$. We use this approach to prove facts about $F$ and then we extend certain results about $F$ to results about $f$. In particular, we take this approach to generalize to $D^n$ the classic Schwarz Lemma on the disc $D$ and give sufficient conditions for a bounded analytic function on $D^n$ to be uniquely determined by its values on a finite set of points. In terms of the Pick problem on $D^n$, we give sufficient conditions for a Pick problem to have a unique solution.

Advisor: Jim Agler

### February 18, 2011

### 10:00 AM

### AP&M 7218

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