Bochner Schwartz type theorem for conditionally positive definite Fourier hyperfunctions (Q1417886)

The Bochner-Schwartz theorem, which renders positive functions as Fourier transforms of positive measures, can be generalised to many classes of ordinary and generalised functions, in particular Fourier hyperfunctions. Here, the authors show such a representation theorem in the case where the standard condition of positivity is replaced by conditional positivity of order \(s\), i.e., \[ \langle u,(D\varphi)\ast(D\varphi)^\ast\rangle \geq 0 \] holds for the Fourier hyperfunction \(u\), any test function \(\varphi\) and differential operator \(D\) with purely homogeneous symbol of order \(s\).