Expo-rational B-splines Expo-rational B-splines Expo-rational B-splines

A new type of B-spline the expo-rational B-spline is introduced. The heuristic motivation for its introduction comes from important similarities in several celebrated mathematical constructions originating in approximation theory, differential geometry and operator theory. The main result of the paper is the derivation of an Edgeworth and a steepest-descent/saddlepoint asymptotic expansion which shows that the expo-rational B-splines are the asymptotic limits of polynomial B-splines when the degree of the latter (or, equivalently, the number of the knots of the latter) tends to infinity. We show that, as a consequence of their nature as asymptotic limits, the new B-splines exhibit ’superproperties’ by outperforming usual B-splines in a number of important aspects: for example, in constructing a minimally supported C∞-smooth partition of unity over triangulated polygonal domains of any dimension. We illustrate this ’superperformance’ by 2D and 3D graphical visualization, and discuss ’the price to pay’ for it in terms of computational challenges, and how to deal with them. Finally, we present a first, non-exhaustive, list of potential applications of the new expo-rational B-spline. Dedicated to the 60-th Anniversary of Professor Tom Lyche. Mathematical Subject Classification (AMS 2000): Primary: 65D07; Secondary: 26A24, 26B35, 30B50, 30E20, 33F05, 41A05, 41A15, 41A20, 41A21, 41A30, 41A55, 41A58, 41A60, 41A63, 42A99, 42C40, 44A15, 46E35, 46E40, 46E50, 47A10, 47B06, 47B07, 47B10, 53A04, 53A05, 57R05, 57R10, 57R50, 58A05, 65B05, 65D05, 65D10, 65D17, 65D18, 65F50, 65L60, 65M60, 65N30, 65N50, 65T60, 65Y99, 68N19, 68Q25, 68U05, 68U07