Conditionally positive definite hyperfunctions (Q2574362)

Conditionally positive definite hyperfunctions are defined in accordance with the positive definite hyperfunctions as follows. A hyperfunction \(u\) is said to be conditionally positive definite of order \(s\) if there exists a Gauss transform \(U(x,t)\) of \(u\) such that \(P\overline P U(.,t)\) is positive definite for all linear differential operators \(P\) with constant coefficients of order \(s\) and for each \(t>0\). The main result of the present paper asserts that the following conditions are equivalent for a hyperfunction \(u\): (i) \(u\) is conditionally positive definite of order \(s\). (ii) \(u\) is a conditionally positive definite Fourier hyperfunction of order \(s\). The authors compare the obtained results with known ones.