Journal of The Australian Mathematical Society, 2002
The spherical functions of triangle buildings can be described in terms of certain two-dimensiona... more The spherical functions of triangle buildings can be described in terms of certain two-dimensional orthog- onal polynomials on Steiner's hypocycloid which are closely related to Hall-Littlewood polynomials. They lead to a one-parameter family of two-dimensional polynomial hypergroups. In this paper we investigate isotropic random walks on the vertex sets of triangle buildings in terms of their projections to these hypergroups.
A frequently used method to handle scattered data ap- proximation problems on dierent structures ... more A frequently used method to handle scattered data ap- proximation problems on dierent structures is the interpolation by linear combinations of a single positive denite kernel. Here the un- derlying positive denite kernel is responsible for the quality of the common estimates of the two main issues in this approximation pro- cess. Firstly the approximation error and secondly the stability of the system. In this paper it is shown that we can not ask the kernel for arbitrary good estimates for the attainable error and the condi- tion number of the associated interpolation matrix at the same time. Hereby we use a generalization of Schaback's trade-o principle for RBF interpolation in Rd (R. Schaback. Error estimates and condi- tion numbers for radial basis function interpolation. Adv. Comput. Math., 3, 251{264, 1995.) to more general settings and a wider class of basis functions.
Journal of The Australian Mathematical Society, 2002
The spherical functions of triangle buildings can be described in terms of certain two-dimensiona... more The spherical functions of triangle buildings can be described in terms of certain two-dimensional orthog- onal polynomials on Steiner's hypocycloid which are closely related to Hall-Littlewood polynomials. They lead to a one-parameter family of two-dimensional polynomial hypergroups. In this paper we investigate isotropic random walks on the vertex sets of triangle buildings in terms of their projections to these hypergroups.
A frequently used method to handle scattered data ap- proximation problems on dierent structures ... more A frequently used method to handle scattered data ap- proximation problems on dierent structures is the interpolation by linear combinations of a single positive denite kernel. Here the un- derlying positive denite kernel is responsible for the quality of the common estimates of the two main issues in this approximation pro- cess. Firstly the approximation error and secondly the stability of the system. In this paper it is shown that we can not ask the kernel for arbitrary good estimates for the attainable error and the condi- tion number of the associated interpolation matrix at the same time. Hereby we use a generalization of Schaback's trade-o principle for RBF interpolation in Rd (R. Schaback. Error estimates and condi- tion numbers for radial basis function interpolation. Adv. Comput. Math., 3, 251{264, 1995.) to more general settings and a wider class of basis functions.
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