Positive definiteness of a family of functions

Abstract

General necessary conditions on the real parameters α, β, C, D for the function

\({e^{ - \alpha \rho \left( x \right)}}\left( {C\cos \beta \rho \left( x \right) + D\sin \beta \rho \left( x \right)} \right),\)

where ρ is the norm on Rⁿ, to be positive definite on Rⁿ, are obtained. For ρ(x) = ‖x‖_p, a criterion on these parameters is obtained in the following cases: (i) p = 1or p = 2; (ii) 3 < p ≤ ∞ and n = 2.