Closed form representations for the compactly supported radial basis functions of Buhmann, Wendland and Wu

Abstract

The original compactly supported radial basis functions of Wendland (Adv. Comput. Math., 4, 389–396, 1995) and Wu (Adv. Comput. Math., 4, 283–292, 1995) have a polynomial form and are constructed using a two-step dimension walk strategy. Focussing on the Wendland functions, Schaback (Adv. Comput. Math., 34(1), 67–81, 2011) proposed a one-step dimension walk which is shown to recover the original Wendland functions at every second step but also introduces new examples, the so-called missing Wendland functions at the intermediate steps. In a recent paper (Science China Mathematics Published online, 2025), the analogue of Schaback’s work is presented for the Wu functions and so delivers the so-called missing Wu functions. The original and missing Wendland functions belong to a much wider class proposed by Buhmann (Math. Comput., 70(233), 307–318, 2001). The classical Buhmann functions, which are related to thin-plate spline radial basis functions, also belong to this much wider class. The theme uniting the classical Buhmann functions and the missing Wendland/Wu functions is that they are non-polynomial, and closed-form expressions are not known for all of them. In this paper, we revisit these functions and show how closed-form representations can be given using direct techniques. The results for the classical Buhmann and Wu functions are new, and the resulting expressions for the missing Wendland functions improve on those given in Hubbert (Adv. Comput. Math., 36, 115–136, 2012) and so their implementation should be more straightforward.