Characterization and construction of radial basis functions

We review characterizations of (conditional) positive definiteness and show how they apply to the theory of radial basis functions. We then give complete proofs for the (conditional) positive definiteness of all practically relevant basis functions. Furthermore, we show how some of these characterizations may lead to construction tools for positive definite functions. Finally, we give new construction techniques based on discrete methods which lead to non-radial, even nontranslation invariant, local basis functions.

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