Multiply monotone functions for cardinal interpolation

Abstract

Among other things we derive sufficient conditions for a radial basis function φ: R≥ 0 → R that depend on derivatives of φ(√·): R0 → R being completely or multiply monotone, to admit interpolation on an infinite regular lattice if(x) = ∑ j∈Znf(j)x(x-j), x ∈ Rn, to f:Rn → R, where the cardinal function if(x) = ∑ j∈Zncjø(∥x - j∥), x ∈ Rn, satisfies χ(λ) = δ0l for all l ε{lunate} Zn. © 1991.