INTERPOLATION BY HARMONIC POLYNOMIALS
Abstract
Let Hn(u; z) denote the harmonic polynomial of degree at most n found by interpolation in 2n +1 points to a function u given on the boundary C of a region D of the complex z-plane. It is proved that (a) for any bounded D there always exist interpolation points on C so that Hn can be uniquely determined for each n, and (b) for a wide class of Jordan regions D and for boundary data u with a smooth first derivative on C the points of interpolation on C can be chosen so that lim as n approaches infinity Hn(u; z) exists, z epsilon C + D, and gives the solution of the Dirichlet problem for u and D. Explicit formulas are derived for Hn in the case of interpolation on a circle and on an ellipse, and convergence is proved in these cases for arbitrary contnuous boundary data. Various generalizations are indicated.
Document Details
- Document Type
- Technical Report
- Publication Date
- Dec 01, 1961
- Accession Number
- AD0268335
Entities
People
- J. H. Curtiss
Organizations
- University of Miami