For a xed conditionally positive deenite function we consider the linear space R containing all nite linear combinations of translates of augmented by a polynomial part. We equip R with a topology and provide characterizations for the largest space T among all linear spaces whose elements are continuous on R. Our approach serves to prove that all conditionally positive deenite functions are ava...

Two sets of the Heun functions are introduced via integrals. Theorems about expanding functions with respect to these sets are proven. A number of integral and series representations as well as integral equations and asymptotic formulas are obtained for these functions. Some of the coefficients of the series are orthogonal (J-orthogonal) functions of discrete variables and may be interpreted as...

For n + 1 points in IR, in general position, the Kergin polynomial interpolant of C functions may be extended to an interpolant of C functions. This results in an explicit set of reduced Kergin functionals naturally stratified by their dependence on certain directional derivatives of order k, 0 ≤ k ≤ d − 1. We show that the polynomials dual to the functionals depending on derivatives of order k...