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G. I. Olive, Karpacz Lectures 1975. 6. E. Lukacs, Characteristic Functions, (London 1970). 7. T. Hida, Stationary Stochastic Processes, 1970) . (Princeton 42 8. B. Simon, The P(~)2 Euclidean Quantum field Theory, (Princeton) . 9. J. Frohlich, Helv. Phys. Acta 47, 265 (1974) and Adv. Math. (to appear). 10. C. R. Klauder, Corom. Math. Phys. 16, 329 (1970). 11. R. Klauder, A Characteristic Glimpse of the Renormalization Group, (Bell Labs. preprint 1975, to appear in J. Math. ). 12. L. Streit, Acta Phys.

88, 588 (1975). 14. J. Eachus, L. Streit, Rep. Math. Phys. 4, 161 (1973). 15. T. Hida, L. Streit (in preparation) . 16. G. Sudarshan 17. R. Klauder, L. Streit, W. Wyss, unpublished. 18. Discussions with V. Enss were very helpful in clarifying this point. 19. C. Hegerfeldt, Contribution to the ZiF-Symposium Dec. 75, and K. C. Hegerfeldt, Is the Wick Square Infinitely Divisible 7, ZiF preprint,1976. 20. C. Hegerfeldt, Corom. Math. Phys. 45, 133 (1975). 21. K. Osterwalder and R. Seneor, The S-Matrix is Nontrivial for Weakly Coupled P(~)2 Models, preprint 1975.

X N) is a real polynomially bounded functN ion on R • When X is a polynomial we shall also write P(fl, ... •• ,N explicitly and X £ P(SR). For polynomials we have by [50J that I (P) :: 5 (P (f 1 ' ••• , fN) ) 58 is well defined. Now I is extended to E by a method due to M. Riesz and M. Krein used in the classical moment N problem on R • Choose X E and let P be a polynomial £ and define ext I(P + AX) =,I(P) + A c for A real; with c to be chosen so that the extension is positive. Let Y, Z be polynomials so that Z(w) Then Y~) - Z(w) = (Y-X) ~) + (X-Z) ICY).