By Carlos A Berenstein

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Def. 3,1). ,0,nt). Since is to prove that, and a def = n~ ~j, j=l the spaces ~w and ~g we shall usually replace each ~ in ~c by its modification which will be called w. ,Qn arbitrary non-negative then for + p(o). ,[ Moreover, n) for Q as above, = ~ j-1 Pj(I[jl) the function ~(~) = Q(I~I) is in K . Theorem 2. Proof. For each ~ c T~c, ~ is a PLAU-space. 3 First we have to verify condition w = Zmj~ (vii) of Def. 3,I. ~ provided we can show that: (a) each < is nontrivial; ] ] and,(b) the AU- and BAU-structures of the spaces 9 T .

C6exp(-~(~)) (CC;I)~ s ~(C,X,{rj},{aj}). c. topology ~(~) on ~ 32 having for the basis of neighborhoods of the origin the system of all sets ~ of the form (5). Topology % ( % ) . Let {Hs}s>l be any concave sequence of positive Hs/S ÷ 0. Fix a positive number ~ and a bounded numbers, H s ~ ~, sequence (~s}s>l of positive numbers. Then the series oo {6) k(~) = k((Hs};(Cs};p;~ ) = [ s=l ~seXp[-(s+p)oJ(~) + Hslrll] is locally uniformly convergent in ~n and defines a majorant in the sense of Chap. I.

0. k(~) Thus, choosing = C- 1 Then (52) the m ~ ~f(@) B C A(m;~) and k as above. by We claim (6) and + 1, for some that (50), ~ ~ s e X p [ ( H s - A ) t~[ - ( s + ~ ) ~ ( ~ ) + ~ ( ~ ) k ( w ( ~ ) ) ] s=l for HSo> I~I _> E, A, and family from so large : I¢1 < E < m* s ~(£) const, m*(~) such that (51), ~A~(~) is a B A U - s t r u c t u r e , a function m(~) E > 0 we obtain > C-I~s mine o for each A = Rs to check. e. 0 Let us choose • m = m({Cn};A;... ). Now let (51) To prove 2 Rs }, o condition k(~)/m(~) Cn = {x : [xl 0 > it suffices that to find If m is given by (49) and {Vj} as above, sequence of points Igjl : ~ and gj,{j s aVj for ~(gj) = Cj+ I- Cj.