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New PDF release: Asymptotics of Analytic Difference Equations

By Geertrui K. Immink (auth.)

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Extra resources for Asymptotics of Analytic Difference Equations

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74) and the estimates found above it follows immediately that A is an inverse of A B. Observe that the definition of A is independent of the values of a I and a2 and the shape of the regions S(R). 5. Conclusions. 4 we can collect the following information concerning the constants c i and b i. If i ~ g, we may take for c i any positive number smaller than the minimum of all constants c (cf. 3, case I) corresponding to blocks B of A with the property that k(B) ={ki}. 59 In the case that i = £, and hence k.

We start from an arbitrary matrix function A ~ M ( S ) Although it will soon become clear that the shape of the regions S(R) is determined by the canonical form A c of A. 3, there exist linear mappings A S(R) possessing the required properties. For g,a,r notational convenience we shall assume that all 'blocks' B are of dimension n. 2. Definition of the regions S(R). The basic ideas of the construction that follows are simple and easily understood. However, the detailed elaboration of these involves a great deal of tedious calculations.

We define a linear mapping ~ on B ~f(s) = A'f(s) g,a,r ~(R)) as follows if s C S'(R), ~f(s)=Af(s)+Yo(S)Yo(Sh+½) -I I'(Sh+~) if s6S(R'). e. 8). Hence it follows that ~ is a right inverse of A B. 56) and the results obtained by the above described method with respect to A and A', one readily verifies that N ~(R) [IAf[[g,a,r+l_k - ~(R) < K Ilf[ g,a,r ' - R >-- I ' provided that (g, lal) < (k,~), where c= I~I min{~h,(gh)', + + eh,(S h) V}. I (g,a) < (k,-b) < O. By a corollary to a theorem of Phragmgn-LindelSf is reduced to {0} when g ~ K .

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