By Stevan Pilipovic
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Extra resources for Asymptotic behaviour and Stieltjes transformation of distributions
1), let f = where f+ g c ±> |x|v+mL( Ix I) and f_ e S I that for every ф g + f (suppf_ <= (-®,0]). 1. implies S f (kx ) < -n----,ф(х) > -► < g ( x ) ^ ( x ) > k vL(k) as к -► ®. ,( х),ф(х) > k vL(k) * V+1 as к -► ®. 3. Let f + g F2
8. 11) f (x ) ^ X v L 2(X) as x -► ® for V > 0, where L 2 is slowly varying function at infinity. 9), has quasiasymptotic behaviour of order V related to x vL 2 (x). Proof. 12) tends to zero as к + •. So we have to prove that — ----- kV kU if(kx) (x)dx -► fxvL 0 (x)ф ( х М х as к -*■ <*>. 11) we have that for every e > 0 there exists M > 0 such that |f(x) - xvL 2 (x)| ¿ £x v L 2(x ) + M for x £ I. This implies |f(kx) - *(xk)vL 2 (kx) I £ e(xk)vL 2 (kx) + M Thus we have ^ CO for x £ I. 00 ----- [f( к х ) ф ( х Ы х --- ------ f (kx)vL 9(k x H ( x ) d x | й J.
2J at O+ of order (-m,ln(l/k)) related to (1/k) m . 2. e. e. of order (a,L) and of length Z t related to ke ”*L (k) ( ( 1 / k ) 1 / k ) ). 2. e. at » of order (a,L) and of length s and let 0 S Z^ < Z^ < s. Suppose that N f related to " l a i (fLi> «i+1 a t ~ (0+) к“'* l L i i (к? ( (1/k )°**'1L^ ^( 1/k) ) , N [ bS f L i h i + ! f related to at “ (0+ k “ '*2L. (к)1 ((l/k)a+tlL, (1/k)). e. e. at ® of order (a,L) and of length s. The similar conclusion holds for the point O+ as well. 1. e. at ® ( 0 + ) of order (a,L) and of length s.
Asymptotic behaviour and Stieltjes transformation of distributions by Stevan Pilipovic