I'm reading a proof in Eymard's paper about the Fourier Algebra in which he refer to a proposition and a theorem in Roger Godement paper Les Fonctions De Type Positif et la Theorie Des Groupes, which can be found here (just click on view pdf) . is there an english translation for the hole paper?. if not, I'm trying to understand Proposition 12 in page 68 and Theorem 17 in page 73.
Any help would be appreciated.
Thank you.
Transcription of the paragraphs in question:
Page 68: Proposition 12. Toute fonction continue de type positif, sommable pour $dx/(\rho(x))^{1/2}$, est de carré sommable pour $dx$ (c'est à dire: $\mathcal P^1 \subset \mathcal P^2$).
Page 73: Théoreme 17. Toute fonction continue $\phi(x)$, de type positive et de carré sommable, est de la forme $$\phi(x) = \psi * \psi^\sim(x) = \int \psi(xy)\overline{\psi(y)}dy \quad \text{ où }\quad \psi \in \mathcal P^2.$$
Best Answer
I can translate the French, though I'm unfamiliar with the math, so it'll be a very literal translation that hopefully you can interpret:
Prop 12: Every continuous function of positive type that is summable for $dx/(\rho(x))^{1/2}$, is square-summable for $dx$ (that is: $\mathcal{P}^1 \subset \mathcal{P}^2$).
If that means something to you, great! If not, then we'll need a French speaker who's a better mathematician that I to intervene!
Edit given addition to question:
Theorem 17: Every continuous function $\phi(x)$ that is positive and square-summable is of the form...