ABSTRACT Sivakumar Sivakumar will work on problems form the theory of radial-basis functions (RBF). The first part proposes the general study of cardinal interpolation (i.e., interpolation at the multi-integer lattice) by shifts of RBFs. The second part of the proposal deals with the analysis of the cardinal-interpolation operator. Stability estimates relating to scattered-data interpolation constitute the subject of the third part of the proposal. The final section concerns interpolation of scattered data on Euclidean spheres. The focus here is on interpolants which are closely related to RBFs, and the goal is to develop a substantive quantitative theory of such interpolation methods. As evident from the preceding paragraph, the leitmotif of this proposal is the theory of Radial-Basis Functions, a subject which has been vigorously researched over the last 15 years. These functions provide a viable tool for the interpolation (i.e., modeling) of scattered data. Modeling scattered data is as natural as it is inevitable. Data gathered for experiments in many areas of science (e.g., geophysical or meterological data collected over the surface of the earth via satellites or ground stations) invariably come from scattered sites. It is therefore important to have at one's disposal, flexible and useful methods of producing interpolants (i.e., surface fits) to such scattered data. The radial-basis-function method accomplishes precisely that; indeed, its use was pioneered by the geophysicist Rolland Hardy in his studies on topography. The current proposal seeks to undertake theoretical studies connected with radial-basis-function interpolation. Some of these issues, especially those addressed in the third and fourth parts of the proposal, also have some practical significance. These issues concern the so-called quantitative theory of interpolation which strives to explain how stable the interpolation (modeling) process is likely to b e.
