Operator Theoretic Methods in Nevanlinna-Pick Interpolation
| dc.contributor.author | Hamilton, Ryan | |
| dc.date.accessioned | 2009-03-31T19:31:54Z | |
| dc.date.available | 2009-03-31T19:31:54Z | |
| dc.date.issued | 2009-03-31T19:31:54Z | |
| dc.date.submitted | 2009-03-26 | |
| dc.description.abstract | This Master's thesis will develops a modern approach to complex interpolation problems studied by Carath\'{e}odory, Nevanlinna, Pick, and Schur in the early $20^{th}$ century. The fundamental problem to solve is as follows: given complex numbers $z_1,z_2,...,z_N$ of modulus at most $1$ and $w_1,w_2,...,w_N$ additional complex numbers, what is a necessary and sufficiency condition for the existence of an analytic function $f: \mathbb{D} \rightarrow \mathbb{C}$ satisfying $f(z_i) = w_i$ for $1 \leq i \leq N$ and $\vert f(z) \vert \leq 1$ for each $z \in \mathbb{D}$? The key idea is to realize bounded, analytic functions (the algebra $H^\infty$) as the \emph{multiplier algebra} of the Hardy class of analytic functions, and apply dilation theory to this algebra. This operator theoretic approach may then be applied to a wider class of interpolation problems, as well as their matrix-valued equivalents. This also yields a fundamental distance formula for $H^\infty$, which provides motivation for the study of completely isometric representations of certain quotient algebras. Our attention is then turned to a related interpolation problem. Here we require the interpolating function $f$ to satisfy the additional property $f'(0) = 0$. When $z_i =0$ for some $i$, we arrive at a special case of a problem class studied previously. However, when $0$ is not in the interpolating set, a significant degree of complexity is inherited. The dilation theoretic approach employed previously is not effective in this case. A more function theoretic viewpoint is required, with the proof of the main interpolation theorem following from a factorization lemma for the Hardy class of analytic functions. We then apply the theory of completely isometric maps to show that matrix interpolation fails when one imposes this constraint. | en |
| dc.identifier.uri | http://hdl.handle.net/10012/4305 | |
| dc.language.iso | en | en |
| dc.pending | false | en |
| dc.publisher | University of Waterloo | en |
| dc.subject | Operator theory | en |
| dc.subject | Nevanlinna-Pick interpolation | en |
| dc.subject.program | Pure Mathematics | en |
| dc.title | Operator Theoretic Methods in Nevanlinna-Pick Interpolation | en |
| dc.type | Master Thesis | en |
| uws-etd.degree | Master of Mathematics | en |
| uws-etd.degree.department | Pure Mathematics | en |
| uws.peerReviewStatus | Unreviewed | en |
| uws.scholarLevel | Graduate | en |
| uws.typeOfResource | Text | en |