# Asymptotic behaviour and Stieltjes transformation of by Stevan Pilipovic By Stevan Pilipovic

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Additional info for Asymptotic behaviour and Stieltjes transformation of distributions

Example text

Let us only remark that (iv) follows from , T. I, p. 72, Th éo­ rème X. 2. Let f s E' and f ’ g at ±~ related to k vL ( k ) . Then L(k) = I, к > a for some a > 0, and v e -M. 10) can be extended on S. Proof. It is well-known that f can be written in the form f = i r(k). ,m, are continuous functions with compact supports. If F is a continuous function with the compact supports, then one can easily prove that for some,C Iim < --- t — ,ф (x) > = < C 6(x),ф(х) >, ф e s . _ As it is usual, we identify locally integrable functions with the corresponding distributions.

I (l n x ) » G 9 n (x) = Zt ± l 0, X > 0 x < 0 f H0 i (I n x), x > 0 ¿9± [ o, 0. 1 (X) P=I are e q u a l . 1 к 21 = ^ a p C - D p < H 2 1(кх),(хр Ф(х))(р) > + < (G2 д ( к х ) ,ф(х) >. P =1 Functions H 2 1 and G 2 1 are bounded on (-®,0). 6 *) (IG2 Х (кх)I ,|H2 >1(k x)I} < M, x > 1/k, 0 < x < 1/k. 7) Jf2+ ( k x H ( x ) d x / ( k vL(k) ) < ®. 7) holds for every ф e S. If we put f 2 2 (t) = f 2 ('et) ’ t > 0 > by the same argument as above, one can prove that for every ф e S there holds о Jf 2+ ( к х ) ф ( х М х / (kvL(k) ) < ®.

Kg) F(kt ) 1I 1I 1I + 1I , 2 I F(kt ) C J k aL(k) J k° + 0. 4). ,0 ), I2 -* 0 as к + ®. 2. Let f ^ g at » related to к L(x). Then, there is n e ]Nq such that p+a > 0 and D~pf has the asymptotic at » related to ka+pL(k) with the limit c ^a+p+ e » in this case g = Cfa + 0 .