# Analytically Uniform Spaces and their Applications to by Carlos A Berenstein

By Carlos A Berenstein

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Extra resources for Analytically Uniform Spaces and their Applications to Convolution Equations

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H s = h(s) k = k({Hs};{es};~ ) ~ ~(~). the last along 2, let f be a fixed entire for each in ~+, the sequence an arbitrary majorant (38) follows as the first one. e • for all ~ ~ ~n (of. and any > O, (48) with f = \$ 2. tf(~)l C = C 2. for some < c e Blnl-~c°({) The Paley-Wiener ~ ~ ~; and~this theorem (cf. completes Remark 2) then implies the proof of Proposition suitable (Cj }j~l Given ¢(g) To complete the proof BAU-structure for ~'. of positive any {Cj} numbers ~ ~ and = inf(C n - nm({)) of Theorem l,it remains Let ~ be the class such that arbitrary 1 = m(O) A > O, to exhibit a of all sequences < Cj+ 1 C > O, - Cj .

For M c ~ , B(n2+n). supp(~M¢ ) Since 1~0(~31 (27) ! ), f ,n B(2IMII); we obtain <_ CC6, 0 ~ e x p [ - X ~ ( ¢ - t ) where we used the subadditivity Now let us set of w. d = n2+n n+l X=v n 2+n + ~ and, l~(~-t) l l a o ( t ) ] d t <-- CC°,0 e - d ~ ( ¢ ) (28) c and S - ~(t)]dt exp[(~-X)m(¢-t)]dt , 40 where b is the constant from condition C > 0 We also fix (y), Def. i. so that S (29) 2nccv dt (l+jtl)n+l ,0 n2+n ~ ~n Then inclusion (26) is verified for e 2 n +n M = 0. In order to estimate those terms in (26) for which JMJl = s > 0, we have to shift the integration in the convolution A M ~M ¢ from ~n to the variety F~ = {T = t+in ~ cn : ~ = _ ~-[ as~(~_t) ' t ~ ~n }.

Which completes = ~(g). the proof of part 2. ~w(B(s)) : sup I\$(£)le be the set of all lattice points that [M[ ~ n/~. +[Xnl that for a convenient (cf. Remark 3), than ~ ( i n d ) . ~ c for Z ! i, ql > I; Now let ~ be a closed convex neighborhood ~ If For (23) and (24) prove the inclusion c U(C,~,{rs},{as}) ' the topology ~ K i. (17) give (24) 3. from and (23) follows which still has to be checked ~(k) and ~Z = aq . ~+i ! q£ ~ q, and ~ = 0 ! s}. We claim ~(C,~,{rj},{aj}) M = (m I .... ,mn) c~. in ~n such let us set M e ~ , and S0 : {x : IX I < (n+l) By [9] there exists a partition of unity fn}.