ASYMPTOTICS OF REGULAR CONVOLUTION QUOTIENTS

The asymptotic behaviour of a class, of generalized functions, named regular convolution quotients, has been defined and analysed. Some properties of such asymptotics, which can be useful in applications, have been proved.

ple [5], [6], [7] and [8] ), which can be applied in solving a lot of mathematical mo- dels.A distribution T has S-asymptotics related to a positive and measurable fun- ction c iff lim (T*w)(h)/c(h) (S'w)(0) for every w D We write" T s c(h).S, In thls paper we shall enlarge the definition of S-asymptotics of distribu- tions to the regular convolution quotients having in view the application of this class of generalized functions.

REGULAR CONVOLUTION QUOTIENTS.
By Boehme [I] an a p p r o x i m a t e functions (u n) c L(R) satisfying the following conditions" i) 5 R Un(X) dx nN ii) there is an M0 such that R lu(x) dxM iii) there exists a sequence (kn)CR + such that k n supp u nc[-kn,kn neN identity nN; is a sequence of -0 as n and / will be the set of all approximate identities and (Un)e u ng C , n N}.A d e f i n i n g s e q u e n c e for a regular convolution quotient is a se- quence of pairs ((fn,Un)), where (fn) C Lloc(R), (Un)e and for all m,ngN the folio- wing convolution products are equal- iv) fn*Um fm*Un the asterisk is the sign of the convolution).
Two defining sequences ((fn,Un)) v) f *v n m gm*Un for n,mN and ((gn'Vn) are said to be equivalent if" By fn/U n we shall denote the equivalence class containing the difining sequence ((fn,Un)).A r e g u 1 a r c o n v o 1 u t i o n q u o t i e n t X is an equi- valence class of defining sequences The regular convolution quotients are a vector space when the usual multiplication by scalars and addition of fractions is used; we denote it by B(Lloe, ).The space B(oe, A contains D' (space of Schwartz's di- stributions) under the isomorphism: D'9 T -(T*Vn)/V n B(Lloe, ), where (Vn)E Moreover, B(Lloc, contains the class of all regular Mikusinski operators.Both of these containments are proper. Let (h n) be any continuously differentiable approximate identity.By D n/hn e B(Lloc,6 is defined the differentiation operator.The derivative of an X fn/hn % & B(LIo c i) is now, defined to be DX (fn hn)/(Un*hn) & S(hoc')" For a distribution T e D' and w&D we shall write T(w) < t,w > We shall use the following properties of elements belonging to and distributions defined by local integrable functions I.For (fn)Lloc and (Vn) e we have <fn(x+h),%(x)> (fn*Vn)(h) where n (x) v n(-x) 2. If (u n) and (v n) belong to , then (Un*Vn) e as well.
3. If (fn*Vn)(O) 0 ,neN for every (v n) e Zi then fn (x) 0 for almost all x R 3. S-ASYMPTOTICS OF REGULAR CONVOLUTION QUOTIENTS Let E be the set of all real valued, positive and measurable functions: R -+ DEFINITION i.A regular convolution quotient X has S-asymptotics at infinity, related to c e r. and with the limit U Fn/U n E B(oc,) if there exists ((fn,Un)) )/u n belongs to B(Lloc,)because of (S*Un)*U m (S*Um)*U n for every m,n X s, c(h).(S.Un)/Un h-.Let u'remark that (S*Un)/U n corPesponds to Se by the mentioned isomorphism.In such a way S-asymptotics of regular convolu- (fn*h)/(Un*hn) where (h n) is any continuously differentiable approximate identity.Now, the following relation is true" fn*h )*v n (h) fn* (h*vn) )(

Hence
, /(Un,hn) s fn hn) c(h).(Fn*h)/(Un*hn) and DX s, c(h).DU, h- where U Fn/U n This proposition can be useful in applying regular convolution quotients to dif- ferential equations.The next proposition precises the analytical form of the function c g E which measures the asymptotical behaviour of a regular convolution quotient and the form of the regular convolution quotient U, the limit in Definition I. PROPOSITION 3. Suppose hat X B(Lloc, and X s c(h).U h-where c. and U Fn/U n If F n 0 for one n then c(h) exp(ah) L(exph) Now, the proof of Proposition 3 follows directly from Proposition 4.3 in [5] or propositions 9 and 10 in [7].PROPOSITION 4. If X B(Lloc,), then X has a compact support if and only if X ,s c(h).O lhl-> for any eel Proof.We know (see [10]) that X e B(Lloc,) has compact support if and only if there is a (Un)e such that UnX fn n e N and fn 'neN has compact support Moreover, if X has compact support, then this is true for every gn XJn ne N ((gn,Jn)) X Suppose that supp fn C I-an'an] and supp v n c [-kn,kn], an> 0 kn> 0, every (v n) e Zi We shall write X ,s c(h).U h -This definition does not depend on the defining sequence ((fn,Un)) in the equi- valence class X Let ((gn,Jn)) e fn/Un and let Gn/J n & B(Lloc,/i) vn) (0) nS and (v n) e Then ((Fn,un)) and ((Gn,Jn)) belong to the same class because of" < (Fn*Jm) ,n> ((Fn*Jm)*Vn)Us, % > for every (Vn) and m,n,N Hence, Fn*Jm Gin* u n for m,n S PROPOSITION I.If a distribution T has S-asymptotics, T c(h).S, h-c 7.then the regular convolution quotient X (T*Un)/U n which corresponds to has S-asymptotics, as well and X s c(h).(S,Un)/Unh -Proof.For every v n) e we have" ((T*Un)*Vn) (h) (T* (Un*Vn) )

(
>0 and F n(x) C n exp(ax) where a{R C neB C n 0 varying func tion.and L is a slowly Proof.L is a slowly varying function by definition iff LeE and lim L(ux)!For slowly varying functions see, for example [9] )-By Definition there exists ((fn,Un)) e X such that (fn*Vn)(h) /i h -lim c(h) (Fn*Vn)(0) ne N for every (Vn)e +k Hence neN and (v n) e'.Then we have'(fn*Vn)(h):0 for hl>an n fn