Bochner–Schwartz Theorems for Ultradistributions

AbstractWe prove the Bochner–Schwartz theorem for the ultradistributions in the quasi-analytic case. In other words, every positive definite ultradistribution of class {Mp} is the Fourier transform of a positive {M}-tempered measure μ, that is, ∫exp[−M(ε|x|)]dμ<∞ for every ε>0, whereM(t) is the associated function ofMp. To prove this, we show that every positive elementuin S′{Mp}is a positive {M}-tempered measure, and that every positive definite ultradistribution of Roumieu type is nothing but a positive definite element in (SMpMp)′ and hence is the Fourier transform of a positive {M}-tempered measure. Our result includes the cases for Roumieu type and Beurling type and also both for all the non-quasianalytic cases and most of the quasi-analytic cases