Let m be a nonnegative integer, {z₁,,z_n} be a sequence of distinct points in open unit disk D of complex plane and {w₁,,w_n} be an arbitrary sequence of complex numbers. The generalized Nevanlinna-Pick problem on Hardy Space H² of D is to find a condition that determines whether there exists a function f in H² 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...[ Read more ]

Let m be a nonnegative integer, {z₁,...,z_n} be a sequence of distinct points in open unit disk D of complex plane and {w₁,...,w_n} be an arbitrary sequence of complex numbers. The generalized Nevanlinna-Pick problem on Hardy Space H² of D is to find a condition that determines whether there exists a function f in H² 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 H².