Conditionally Positive Definite Functions and Their Application to Multivariate Interpolations Original Research Article

We derive integral representations for conditionally positive definite functions of various types. Using these representations, we show that certain symmetric linear combinations of translates of conditionally positive definite functions are positive definite functions and can be employed in multivariate scattered data interpolation. We also obtain a necessary and sufficient condition for the smoothness of conditionally positive definite functions. As a corollary, we establish a generalization of a theorem of von Neumann and Schoenberg on integral representation for functions of negative type which play a central role in isometric imbedding theory and radial basis function interpolation.