Interpolation with symmetric polynomials
Interpolation problems in more than one dimensional ambient spaces are studied in this article. Whenever the dimension of the ambient space is more than one, these problems are a lot more difficult to analyse than the classical univariate problems, for instance because not only convergence properties but also existence questions of interpolants, dimensions of the underlying approximation spaces etc.\ now depend on the geometry of the data points. The current article addresses this fundamental problem by observing that certain symmetry properties facilitate the aforementioned problems substantially. They begin with studying even univariate symmetric (even and odd) polynomial interpolation problems (which provide the basis of the then ongoing approach for higher dimensions) and then continue by analysing symmetric (symmetric about a line in two dimensions) bivariate interpolation tasks, thereby reducing the problem substantially and concretely halving the dimension of the approximation space. The authors consider also more general interpolation problems by bringing them back to the previous shape already studied by decomposition into their symmetric and antisymmetric parts.
scientific article; zbMATH DE number 6674007
scientific article