THESIS
2016
Abstract
Let m be a nonnegative integer, {z
_{1},,z
_{n}} be a sequence of distinct points in
open unit disk D of complex plane and {w
_{1},,w
_{n}} be an arbitrary sequence of
complex numbers. The generalized Nevanlinna-Pick problem on Hardy Space H
^{2}
of D is to find a condition that determines whether there exists a function f in
H
^{2} satisfying ❘❘f❘❘
_{2} ≤ 1 and f
^{(m)}(z
_{i}) = w
_{i} for each i = 1,,n. In this thesis, the
necessary and sufficient conditions for this generalized Nevanlinna-Pick problem
are obtained by giving a constructive proof. Also, a formula for the minimal
norm among all the solutions and a form for such function are obtained. With
the utilization of minimal norm function, the analog of usual consequences in
the classical Nevanlinna-Pick problem can be deduced. This thesis also discu...[
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Let m be a nonnegative integer, {z
_{1},...,z
_{n}} be a sequence of distinct points in
open unit disk D of complex plane and {w
_{1},...,w
_{n}} be an arbitrary sequence of
complex numbers. The generalized Nevanlinna-Pick problem on Hardy Space H
^{2}
of D is to find a condition that determines whether there exists a function f in
H
^{2} satisfying ❘❘f❘❘
_{2} ≤ 1 and f
^{(m)}(z
_{i}) = w
_{i} for each i = 1,...,n. In this thesis, the
necessary and sufficient conditions for this generalized Nevanlinna-Pick problem
are obtained by giving a constructive proof. Also, a formula for the minimal
norm among all the solutions and a form for such function are obtained. With
the utilization of minimal norm function, the analog of usual consequences in
the classical Nevanlinna-Pick problem can be deduced. This thesis also discusses
another generalized Nevanlinna-Pick problem, which would lead to a possibility
of solving the Nevanlinna-Pick problem with boundary data on H
^{2}.
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