56,061 research outputs found
Constrained optimization in classes of analytic functions with prescribed pointwise values
We consider an overdetermined problem for Laplace equation on a disk with
partial boundary data where additional pointwise data inside the disk have to
be taken into account. After reformulation, this ill-posed problem reduces to a
bounded extremal problem of best norm-constrained approximation of partial L2
boundary data by traces of holomorphic functions which satisfy given pointwise
interpolation conditions. The problem of best norm-constrained approximation of
a given L2 function on a subset of the circle by the trace of a H2 function has
been considered in [Baratchart \& Leblond, 1998]. In the present work, we
extend such a formulation to the case where the additional interpolation
conditions are imposed. We also obtain some new results that can be applied to
the original problem: we carry out stability analysis and propose a novel
method of evaluation of the approximation and blow-up rates of the solution in
terms of a Lagrange parameter leading to a highly-efficient computational
algorithm for solving the problem
On the Convergence of Kergin and Hakopian Interpolants at Leja Sequences for the Disk
We prove that Kergin interpolation polynomials and Hakopian interpolation
polynomials at the points of a Leja sequence for the unit disk of a
sufficiently smooth function in a neighbourhood of converge uniformly
to on . Moreover, when is on , all the derivatives of
the interpolation polynomials converge uniformly to the corresponding
derivatives of
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