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By Iwaniec H., Kowalski E.

This booklet indicates the scope of analytic quantity idea either in classical and moderb path. There are not any department kines, in reality our purpose is to illustrate, partic ularly for rookies, the interesting numerous interrelations.

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C~Au n u 0 E ~/ . Proof. Let with A . [Ek] Then, be the left resolution of the identity associated since zero is not an eigenvalue III. 2 of A , the function 35 g(~) = d is continuous at g(k) = llEku011 . function 0 < c~ < ~ k=0 with Thus 2 Ek= E k , we have Since g(llAll+a) : flu011 for all f(k) = 1 - (Zk 2 g(O)=0 . a > 0 . is monotonically decreasing. , we can find an a0 > 0 The Since such that for all 5 > 0 with 5 < l~tl + a0 , we have q(5) : maxClf(~)I For every that nO ¢ > 0 there is a g(50) = llEs0u01] < g/2 .

I I It parallels the proof of Notice that equation (17) remains unchanged so that Instead of (22) we obtain (Re) 2 llAZn+lll 2 = llAZnll 2 - 72(2-72) which shows that the sequence 0 q = (l-B2) 1/2 < 1 .

Let Fr,~ v = V~u in (7) to obtain: Re K ~llWlvIL~ (v,v) K (v,v) K (6) ~e have II~IK ~ ~ • Thus, fRe(~T'iv,v>K : v ¢ F(A,K), v / 0) k = inf~ (v,v)K = ~l ~i (n) Theorem. Let the operators A, K, L, and W be as given above. If is a real number such that (12) 0 < ~l < 2k~l ' if x 0 ~ D(A) is an arbitrary initial approximation to the solution x* of (1), and if Az 0 = Lx0-Y' , then the sequence mined by III. 3 (xn] deter- 46 Xn+ I = x n - ~itnZn , n=0,1,2,... , (13) where (AZn,KZn> tn = Re(Lzn,KZn ) , n:0,1,2,...

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