Asymptotics of Analytic Difference Equations by Geertrui K. Immink (auth.)

By Geertrui K. Immink (auth.)

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Suppose t h a t property R' i s so l a r g e that this is true for all hE { 1 , . , N - 1 } that k ~ ( ~ h ) C Zk(A) f o r some k C k l ( A ) ( O b v i o u s l y , R' d e p e n d s on R ) We now d e f i n e ~(R) = S'(R) U S(R'). a c l o s e d r e g i o n ~(R) by with the 40 It is easily seen that the regions ~(R) form another set of S-proper regions. They differ in many ways from the regions S(R) defined above. Thus, for instance, X%~(R)) analogue of is usually not convex for any % >0. 14) still holds in those cases that are of interest here.

57) 49 However, in contrast to the preceding cases, the mapping A is not a left inverse of ~ . ~S(R). 58) O ! 7) is satisfied and let {S'(R), R > 0 } be a set of S'-proper regions of the same type as the regions S(R). We wish to prove that, in the case considered here, the statement made on p. 3 , is true. This requires a slight modification of the S-proper regions and the mappings A. Instead of S(R) we now consider the regions ~(R). 28) hold and, accordingly, ~(R) has the form ~ ( R ) = S ' ( R ) U S(R'), R>O, where R' is a positive number, depending on R.

48)we obtain ll(s) l < K s' C{s) I~ IReyIP(log I~'l)f(~)dCl , s e S(R), R_>|. As path of integration C(s) we take the half-line from s' to infinity with directional angle args. Observing that If(~)I <_ llflI I~I r and putting 52 ~=xs' , x 6 (I,00), we find o~ ll(s)! < K IIfll Is'lr+1 f xr-ReyIp(- log x)Idx. 1 if follows immediately that S(R) IIlllo,o,r+I < K llfIl, R>I, provided r < - ro, where r = - R e y + I. O The linear mapping A has the same properties in this case as in the foregoing one.

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