By Stevan Pilipovic
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Additional info for Asymptotic behaviour and Stieltjes transformation of distributions
P Let us put f 2 (X) J f 2 (x)’ X > 0 L o, X < 0 H9 Л х ) = , . r G ? 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) ) < ®.
1 a > 0, then g is a continuous function and g = Cf , . a+e It can happen that a regular distribution has quasiasymptotic at O+ but not the asymptotic. Such an example is the distribution (2+sinz). s r2+sin(1/ t ), t > 0 * + \o , t < 0‘ We shall show only that for every ф e S(]R) Iim < (sin(k/x)),,ф(х) > = 0 , к-м» hence (2+sin( 1/t) )+ ^ 2 at O+ related to (1 /k )^ = I. OO CO < ( s i n ( k / x ) ) + ,ф(х ) > = J s i n C k x H (x )d x = к J s i n ( l / t ) ф ( k t ) d t = OO CO ® 0« = к J J ( S i n u ) ^ i ( ф(k t ) + (k t ) ф' ( k t ) ) d t = J\j>k ( d t ) .
2. If T = e(x)iax with а ф O t then T ^ ¿6 in S ’ as x->®. If T = 6(m) + x " U + 0 ) with Я ,m e ]N and 0 < ß < I, then T has quasiasymptotic behaviour related to p(k) = k^, p = max(-m-l, -Я-3 ). 2, show that even some con tinuous functions (in some neighbourhood of infinity) without any o r dinary asymptotic behaviour might have the quasiasymptotic behaviour, while those with power asymptotic behaviour do not have necessarily the quasiasymptotic behaviour related to that power function. Another, somewhat non-expected, property of the quasiasymptotic behaviour is its non-coherence with the multiplication with an arbitra ry power function.