asymtotic behaior
TRANSCRIPT
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Chapter I
Asymptotic Behavior of Generalized
Functions
0 Preliminaries
0.1. We denote by Rand Nthe sets of real and natural numbers;
N0=N{0}.The following notation will be used. Ifx= (x1, . . . , xn), y=(y1, . . . , yn) Rn,then xy= x1y1+ + xnyn; x2= x21+ +x2n; x
0
xi
0, i= 1, . . . , n; x
xi
, i=
1, . . . , n; Rn+= {x Rn; x >0}.If x= (x1, . . . , xn) Rn andk= (k1, . . . , kn) Nn0 ,then|k|= k1+ +kn, k! = k1! . . . kn! , xk =xk11 . . . x
knn; D
k = k1/xk11 . . . kn/xknn , f
(k) = Dkf (f(0) = f);
z= (z1, . . . , zn) Cn;D(a, r) denotes the polydisk{z Cn; |ziai|0.
0.2.A conewith vertex at zero in Rn is a non-empty set such that
x andk >0 imply kx .The cone is called solidif int = . Theconjugate cone (dual cone) to the cone is the set{ Rn; x 0for each x}.It is obvious that is also a cone which is convex andclosed. The cone is calledacuteif is a solid cone.
0.3.A function : (a, )R, aR+,is called regularly varying atinfinity[76] if it is positive, measurable, and if there exists a real number
such that for each x >0
limk
(kx)(k)
=x . (0.1)
The numberis called index of regular variation. If= 0,thenis called
slowly varying at infinityand for such a function the letter L will be used.
We then have that any regularly varying function can be written as (x) =
1
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2 Asymptotic Behavior of Generalized Functions
xL(x), x > a,whereLis slowly varying. It is known that the convergence
of (0.1) is uniform on every fixed compact interval [a, b], a < a < b 0,
(i) there exist constants C1, C2 >0 and X > asuch that
C1x L(x)C2x, x X; (0.2)
(ii) limx
xL(x) = +, limx
xL(x) = 0.
We know ([9], p. 16) that if L2(x) , x ,and L1, L2areslowly varying, then L1 L2=L1(L2) is slowly varying, as well. Hence, forx > ,
limh
L(x + h)
L(h) = lim
u
L(log ut)
L(log u)= 1 . (0.3)
The definition of a regular varying function at zero is similar. For the
definition of regularly generalized functions see [156] and [162].
0.4.The class of distributionsf, R,belonging toS+(see0.5.1.)is defined in the following way:
f(t) =
H(t)t1/(), >0,
f(m)+m(t), 0, + m >0,
whereHis the Heaviside function, and the derivativef(m) is taken to be in
the distributional sense (see [192], Chapter I,1). We therefore havef0=,
the Dirac delta distribution; fm=(m), mN; and fp fq=fp+q.Wealso use the notations t+= ( + 1)f+1(t) and t
= (t)+, / N.
Let g S+.We denote g() = f g, R (the is convolutionsymbol).
The sequence mmN C(Rn) is called a -sequence if:a) supp m [m, m], m 0, m ; b) m 0, m N;c) Rn m(t)dt= 1, m N.
If D,then m , m inD,hence{m ; mN}is abounded set inD.
0.5.We will repeat definitions and some basic properties of generalized
functions defined as elements of dual spaces.
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0. Preliminaries 3
0.5.1.The Schwartz spaces of test functions and distributions on Rn are
denoted byD=D(Rn) and D = D(Rn), respectively (Rn will be omittedwherevernis not fixed). The spaceDis a locally convex, barrelled, Monteland complete space. If a filter with a countable basis is weakly convergent
inD,then it is convergent inD with the strong topology, as well (cf.[146]).
Sis thespace of rapidly decreasing functionsand its dualS is thespaceof tempered distributions[146]. For a closed cone Rn, S={fS;supp f}.In the one-dimensional caseS+={f S(R); supp f[0,
)
}.Recall ([189]):
S() ={ C();p< , pN} ,where
p= supx,||p
(1 + x2)p/2|()(x)| .
BySp() is denoted the completion of the setS() with respect to thenorm
p.Note
S() = pN0 Sp() andS
() = pN0 Sp(),where the
intersection and the union have topological meaning. A sequencefnnNinS() converges tof S() if and only if it belongs to someSq() andconverges tof Sq() in the norm ofSq(). The spaceS is isomorphictoS() if is closed convex solid cone ([189], [192]).
E the space of distributions with compact support; it is isomorphic tothe dual space ofE=C(Rn) (cf. [146]).
DLp,1
p
, is the space of smooth functions with all derivatives
belonging to Lp ([146]),DLp DLqifp < q..
Bis a subspace ofB=DL ,defined as follows: .
Bif and only if|()(x)|0 asx for every Nn0 .
DLp ,1< p is the dual space ofDLq ,1 q
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4 Asymptotic Behavior of Generalized Functions
The elements ofKpare called rapidly exponentially decreasing functions.ThenKpis the dual ofKp.
TheconvolutioninD is defined as follows. LetmmNbe a sequenceinD(R2n) such that for every compact set K R2n there exists m0(K)such thatm(x) = 1, x K, mm0(K) and
supxR2n
|()m(x)|< C , Nn0.
The convolution ofT, S D is defined by
T
S,
= lim
mT(x)S(y), m(x, y)(x+ y)
,
D,
if this limit exists for everymm(then, it does not depend onmm).By the BanachSteinhaus theorem we know that T S D.
The spacesD() andS() with the operationare associative andcommutative algebras. The convolution in this case is separately continu-
ous.
We refer also to [63] and [3] for the theory of distributions.
0.5.2 Ultradistribution spaces
We follow the notation and definitions from [79], [81] and [86]. By Mppis denoted a sequence of positive numbers, M0= M1= 1,satisfying some
of the following conditions:
(M.1)M2p Mp1Mp+1, p N;(M.2)Mp/(MqMpq) ABp,0qp, p N;(M.2) Mp+1
ABpMp, p
N0, N0=N
{0
};
(M.3)
q=p+1Mq1/MqApMp/Mp+1, p N;
(M.3)
p=1Mp1/Mp
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6 Asymptotic Behavior of Generalized Functions
S{M}(Rn) S{M} = ind limh0
SMh ,
whereSMh , h >0,is the Banach space of smooth functions on Rwith
finite norm
h() = sup,Nn0
h||+||
M||M||(1 + |x|2)||/2()L.
The Fourier transform is an isomorphism ofS ontoS;D =D(Rn) isdense in S, S is dense in DL1and the inclusion mappings are continuous.
The strong dual ofS, S,is the space of tempered ultradistributions(of Beurling and Roumieu types). There holdsD
S
E
,where
means that the left space is dense in the right one and that the inclusion
mapping is continuous. Thus,E S D.We denoteS+={fS(R); supp f [0, )}.
Let
S[0,)={C[0, ); =|[0,)for some S}with the induced convergence structure fromS; its strong dual is in factS+.
An operator of the form: P(D) =
||=0
aD, a C, Nn0 ,is
called ultradifferential operatorof class (Mp) (of class{Mp}) if there areconstants L >0 and C >0 (for every L >0 there is C >0) such that
|a|C L||/M||,||N0.0.5.3. Fourier hyperfunctions.There are many equivalent definitions
of hyperfunctions, Laplace hyperfunctions and Fourier hyperfunctions (cf.[71], [80], [144], [145], [204]), but we will use definitions and results collected
in [75]. Let Ibe a convex neighborhood of zero inRn and let be a non-
negative constant. A functionF,holomorphic onRn+iI,is said todecrease
exponentiallywith type (), 0,if for every compact subset K Iand every >0,there exists CK, >0 such that
|F(z)| CK,exp(( )|Rez|), zRn + iK . (0.7)The set of all such functions is denoted by
O(Dn + iI),(
O(Dn + iI) for
= 0),where Dn denotes the directional compactification ofRn : Dn =
Rn Sn1 (Sn1 consists of all points at infinity in all direction). SpacePis defined by
P= ind limI0
ind lim0
O(Dn + iI) . (0.8)
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0. Preliminaries 7
The dual space ofP is the spaceQ(Dn) of the Fourier hyperfunctions.Q(Dn) is a space of F S-type. We can give a representation of elementsfromQ(D
n
).LetObe the sheaf of holomorphic functions. DenoteUj= (D
n + iI){Im zj= 0}, j= 1, . . . , n; (Dn + iI)#Dn =U1 Un ,
(Dn + iI)#j Dn =U1 Uj1 Uj+1 Un.
Then
Q(Dn) =O((Dn + iI)#Dn)/n
j=1O((Dn + iI)#j Dn) . (0.9)
Thus, f Q(Dn) is defined as the class [F],whereFO((Dn + iI)#Dn).Fis called a defining functionoffand it is represented by 2n functions
F, F=F,where FO(Dn +iI); Dn + iI=Dn + i(I) is aninfinitesimal wedge of type Rn + i0,are open -th orthants in R
n.
An f L1loc(Rn) is called a function of infra-exponential typeif forevery >0,there exists C >0 such that|f(x)| Cexp(|x|), x Rn.Then by fis denoted the hyperfunction defined by f.
We denote by the set ofn-th variations of{1, 1}.The boundary-value representation off Q(Dn) is:
f= [F] :=
F(x +i0) . (0.10)
F(x + i0) denotes the element of the quotient space given in (0.9); it is
determined by F.
The dual pairing between Pand g= [G] Q(Dn
) is given by
g, =Rn
g(x)(x)dx=
Imw=v
G(w)(w)dw, w=u + iv,
wherev I.Similarly,Q(Dn), >0,is defined usingO instead ofO(cf.
Definition 8.2.5 in [75]).
An infinite-order differential operatorJ(D) of the form
J(D) =|a|0
baDa with lim
|a|
|a|
|ba|a! = 0, ba C ,
is called a local operator. J(D) maps continuouslyO(U) intoO(U), Ubeing an open set in Cn,and alsoQ(Dn) intoQ(Dn).
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8 Asymptotic Behavior of Generalized Functions
TheFourier transformonQ(Dn) is defined by using the functions= 1(z1) . . . n(zn),where k=1, k= 1, . . . , n, = (1, . . . , n)and+(t) =e
t
/(1 + et
), (t) = 1/(1 + et
), tR.Letu(x)=
U(x + i0) =
(U)(x + i0) ,
where U Q(Dn +iI), and Udecreasing exponentiallyalong the real axis outside the closed -th orthant. The Fourier transform
ofuis defined by
F(u)()=
F(U)( i0)
=
Imz=y
eiz(U)(z)dz, y I, =+ i,
where F(U) O(Dn iI) andF(U)(z) =O(e|x|) for a suitable >0 along the real axis outside the closed -orthant.Fis an automor-phism ofQ(Dn).
0.6.Let Ebe a locally convex topological vector space. An absolute
convex closed absorbent subset of Eis called a barrel. If every barrel isa neighborhood of zero in E,then Eis called barreled. Throughout this
bookFgstands for a locally convex barrelled complete Hausdorff space ofsmooth functions (the subscriptgstands for general),Fg E= E(Rn),andFgstands for the strong dual space ofFg; observeE Fg. IfT Fgand Fg ,thenT, is the dual pairing betweenTand.We writeF0if all elements ofFgare compactly supported;F0denotes the dual space of
F0; observe that a notion of support for elements of
F0can be defined in
the usual way. InFg,a weakly bounded set is also a strongly bounded one(MackeyBanachSteinhaus theorem). The spaces of distributions, ultra-
distributions, Fourier hyperfunctions, . . . , are of this kind. Furthermore, we
shall always use the notationF=FgifA F, whereA=D, D,orP;in such caseF A =D, D,orQ(Dn),respectively, and we say thatF is a distribution, ultradistribution, or Fourier hyperfunction space, re-spectively. We setF={T F; supp T }.
We suppose that the following operations are well defined onF
g:Differentiation:We assume that /xi,Fg Fg, are continuous oper-
ators. LetkNn0 .Then, k
k1x1. . . knxn
T(x), (x)
=
T(x), (1)|k|
k
k1x1. . . knxn
(x)
, Fg.
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0. Preliminaries 9
Change of variables:IfT Fgand k >0,then by definition
T(kx), (x)
= T(x), 1kn xk, Fg.
IfT Fgandh Rn,T(x + h), (x)=T(x), (x h), Fg.
Furthermore, it is always assumed that k (/k), R+ Fg , and h(+ h), Rn Fg, are continuous. Consequently, by the mean valuetheorem, one readily verifies that both maps are indeed C.
Let E,if is a continuous mapping from Fginto Fg , then wesay thatis amultiplierofFg.ThenTis by definition T, =T,.The set of multipliersofFgis denoted by M().
We shall say that Eis a convolutorofFgif the mapping ,Fg Fg,is well defined and continuous, where(t) =(t). We denotebyM()theset of convolutorsofFg.IfM(),then for T Fg,(T )is defined byT , =T, , Fg .
mmN,a sequence in M() Fg, is called a -sequence inFgifm0, mN,and for every Fg, m inFg, m .
We shall say that the convolutionwith compactly supported elements is
well defined in Fgif a notion of support makes sense in Fgand the followingdefinition applies: givenS, T Fg,where supp S= Kis compact in Rn,their convolution is defined by
S T, =S(x) T(y), (x)(x+ y)= Sx Ty, (x)(x+ y) ,where the function Fghas compact support, supp =K, so that theset K
intKand (x) = 1, x
K.For
F it coincides with the usual
definition of convolution.
We assume thatFgcontains regular elementsf L1loc.They are iden-tified with fitself:
f:f, =Rn
f(x)(x)dx, Fg,
iff L1 for every Fgand ifm0 inFimpliesf, m 0.
There is also another situation in which we shall identify locally in-tegrable functions with elements T F. LetA= D, D,orPandsupposeA F. We identify Twith f L1locifT, = f, ,for all A. In such a case we simply write T= f. For example,T(x) = xex
2
sin(ex2
) S(R) is defined in this way, and not as a regularelement ofS(R) in the sense described above.
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10 Asymptotic Behavior of Generalized Functions
1 S-asymptotics in Fg
1.1 Definition
Definition 1.1.Let be a cone with vertex at zero and letcbe a positive
real-valued function defined on .It is said that T Fghas S-asymptoticbehavior related to cwith limit UifT(x + h)/c(h) converges weakly inFgtoUwhen h ,h ,i.e. (w. lim = weak limit):
w.lim
h,h
T(x + h)/c(h) =U in
Fg (1.1)
or
limh,h
T(x + h)/c(h), (x)=U, , Fg. (1.2)
If (1.1) is satisfied, it is also said thatThas S-asymptotics and we write in
short: T(x + h) s c(h)U(x), h .
Remarks.1) If is a convex cone we could use another limit in .
Let h1, h2.We say that h1 h2if and only ifh1 h2+ ; is nowpartially ordered.
For a real-valued function defined on ,we write
limh, h
(h) =A R
if for any >0 there exists h() such that(h) (A , A + ) whenhh(), h .
If is a convex cone, then the S-asymptotics with respect to this limit
might be defined as:
limh, h
T(x + h)/c(h), (t)=u, , Fg. (1.3)
In case n= 1, the limits (1.2) and (1.3) coincide. We will mostly use
Definition 1.1 in this book, for S-asymptotics defined by (1.3) see also [135].
2) If
Fgis a Montel space, then the strong and the weak topologies in
Fgare equivalent on a bounded set. IfBis a filter with a countable basisand if w.lim
hBT(h) = UinFg,then this limit exists in the sense of the
strong topology. Hence, (1.1) is equivalent to
s. limh,h
T(x + h)/c(h) =Uin Fg.
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1. S-asymptotics inFg 11
(Furthermore, from now on, we will omit the symbol s.for the strong
convergence).
For the first ideas of the S-asymptotics see [3] and [146]. The startingpoint of the theory is [127].
1.2 Characterization of comparison functions and limits
Proposition 1.1. Let be a convex cone. Suppose T Fg has the S-asymptoticsT(x + h)
s
c(h)U(x), h
. IfU
= 0, then:
a) There exists a functiond on such that
limh, h
c(h + h0)/c(h) =d(h0) , for every h0 . (1.4)
b) The limitUsatisfies the equation
U( + h) =d(h)U, h .
Proof.a) Since U= 0,there exists a Fgsuch thatU, = 0.Forthis and a fixed h0
limh,h
c(h+ h0)
c(h)
T(x + (h+ h0))
c(h + h0) , (x)
= limh,h
T((x + h0) + h)
c(h) ,(x)
.
(1.5)
Hence, for every h0
limh,h
c(h+ h0)
c(h) =
U(x + h0),(x)U, =d(h0) .
b) Now we can take in (1.5) any function Fginstead of .Then wehave
d(h0)
U,
=
U(x+ h0),
,
Fg
which proves b).
We now restrict the space of generalized functions to a space of distri-
bution, ultradistribution, or Fourier hyperfunction type. So, we have the
following explicit characterization of the comparison function and limit.
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12 Asymptotic Behavior of Generalized Functions
Proposition 1.2. Let be a convex cone withint =(int is the in-terior of). LetT F have S-asymptoticsT(x+h) sc(h)U(x), h,whereU= 0 andc is a positive function defined onR
n
. Then:a) For everyh0Rn there exists
limh(h0+),h
c(h+ h0)/c(h) =d(h0) .
b) There exists Rn such thatd(x) = exp( x), x Rn.c) There existsC R such thatU(x) =Cexp( x).
Proof.a) Let aint.Then there exists r >0 such that B(a, r).Consequently, for every >0, B(a,r) ,as well.
We shall prove that for every h0 Rn and every R >0 the set (h0+) {x Rn; x> R}is not empty. The first step is to prove that(h0+ ) is not empty.
Suppose thatyB(a,r/2) .Then, for every0>2h0/r >0,
a
(h0+ y)
a
y
+
h0
r ,
henceh0+ yB(a,r) .For a fixed R >0,we can choose such thath0+y> R.Then
h0+ yis a common element for (h0+ ), and{x Rn;x> R}.Now we can use the limit (1.5) when h (h0+ ) , and in the same
way as in the proof of Proposition 1.1 a), we obtain
limh(h
0+),h
c(h+ h0)
c(h)
=d(h0) =U(t+ h0),
U, .
From the existence of this limit, it follows:
1)dextends dto the whole Rn;
2)d(0) = 1; d C(Rn) anddsatisfiesd(h + h0) =d(h)d(h0), h , h0 Rn . (1.6)
We can take h0= (0, . . .0, ti, 0, . . . , ) Rn in (1.6); then the limit
limti0
d(h + h0) d(h)ti
=d(h) limti0
d(h0) d(0)ti
exists and gives
hid(h) =
hid(h)
h=0
d(h), for i= 1, . . . , n .
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1. S-asymptotics inFg 13
We introduce the function Vgiven by
d(h) =e(h)
V(h), i= hi d(h)h=0, i= 1, . . . , n .Then
hiV(h) = 0 for everyi= 1, . . . , n .Consequently,V(h) = 1, h Rn.
For the proof of c), we now haveU(x+h) = exp( h)U(x). Differenti-ating with respect to hand then setting h= 0, we obtain that Usatisfies
the differential equations
xjU= j ,1 j n. Proceeding as in the
proof of b), we obtain U(x) =Cexp( x),for some CR. Remarks.
1. We only assumed thatcis a positive function. But if we know that
there existT FgandU= 0 such that T(x + h) s c(h)U(t), h ,thenwe can find a function c C and with the property
limh,h
c(h)/c(h) = 1.
This function ccan be defined as follows: c(h) =T(x +h),(x)/U,,where is chosen so that U, = 0.In this sense we can suppose, wheneverneeded, that c C, and we do not loose generality.
Similarly, we have (see also Lemma 1.4 in 1.12) that
limh,h
c(h)/c(h+ x) = exp( x) in E
if is a convex cone, int
=
,
( is compact int ).
2. If in Proposition 1.2 we replaceF for the general spaceFg, thena) and b) still hold. On the other hand, c) will not be longer true, in
general. We will show this fact in Remark 5 below by constructing explicit
counterexamples.
3. In the one-dimensional case the cone can be only R, R+or R.
In all these three cases int =.Consequently,dfrom Proposition 1.2 hasalways the formd(x) = exp(x),whereR.
Let us write c(x) = L(ex) exp(x), x R.We will show that Lis aslowly varying function. Proposition 1.2 a) gives us the existence of the
limit
limh,h
L(exp(h + h0))/L(exp(h)) = 1, h0R .
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14 Asymptotic Behavior of Generalized Functions
If =R+,then
limxR+,x
L(xp)/L(x) = 1, p
R+
and this defines a slowly varying function (cf. 0.3). Thus if T Fg(R)and T(x+h)
s c(h)U(x),in R+,with U= 0,then it follows that chasthe formc(x) = exp(x)L(exp(x)), xa >0,where Lis a slowly varyingfunction at infinity. Similarly, if = R, then Lis slowly varying at the
origin, while if = R, then Lis slowly varying at both infinity and the
origin.
4. The explicit form of the function cgiven in 3. is not known inthe n-dimensional case, n2. This problem is related to the extensionof the definition of a regularly varying function to the multi-dimensional
case ([135], [162]) and with certain q-admissibleand q-strictly admissible
functions([192]).
5. As mentioned before, c) in Proposition 1.2 does not have to hold for
a general spaceFg. From the proof of Proposition 1.2, we can still obtainthe weaker conclusion U(x+h) = exp(
h)U(x), h
Rn, which in turn
implies the differential equations (/xj)U= j .For distribution, ultra-distribution, and Fourier hyperfunction spaces, these differential equations
imply thatUmust have the form c) of Proposition 1.2. However, the latter
fact is not true in general. We provide two related counterexamples below.
Let
A0(R) ={ C(R); limx
(m)(x) exists and is finite, mN0} ,
it is a Frechet space with seminorms:
k() = supx(,k],mk
|(m)(x)| .
Let us first observe that the definition ofA0(R) does not tell all the trueabout its elements. Notice that if A0(R), then for m= 1, 2, . . . , wehave limx (m)(x) = 0, while limx (x) may not be zero. Indeed,
the proof is easy, it is enough for m= 1, if limx (x) =M, then
(x) =(0) + x0
(t)dt(0) + M x , x ,
but since has limit at, then M= 0. So, we haveA0(R) = { C(R); lim
x(x) exists and lim
x(m)(x) = 0, m N} .
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1. S-asymptotics inFg 15
Its dual spaceA0(R) contains a generalized function concentrated atwhich contradicts c) of Proposition 1.2. Define the Dirac delta concentrated
atby, := lim
x(x), A0(R) .
Notice that the constant multiples ofare the only elements ofA0(R)satisfying the differential equation U = 0. This generalized function is
translation invariant, i.e., ( +h) = , for all hR; in particular,it has the S-asymptotics
(x + h) s
(x), h
=R .
Therefore, we have found an example of a non-constant limit for S-
asymptotics related to the constant functionc(h) = 1.
We can go beyond the previous example and give a counterexample
for the failure of conclusion c) of Proposition 1.2 with a general cand S-
asymptotics inFg. Letc(h) = exp(h)L(exp h), where Lis slowly varyingat infinity and R. We assume that cis C and, for all m N,c(m)(h)
mc(h), h
(otherwise replace cby cgiven in Remark 1).
Next, we define
Ac(R) ={ C(R); limx
(m)(x)
c(x) exists and is finite, mN0} .
It is a Frechet space with seminorms:
k,c() = supx(,k],mk
|(m)(x)|c(x) .
Note thatAc(R) =c(x) A0(R), consequently,(x) Cc(x), x ,
where C= limx (x)/c(x). An inductive argument shows thatfor all m, (m)(x)(1)mmCc(x), x . Set now gc,=(1/c(x)) Ac(R), a generalized function concentrated at infinityand given by
gc,,
= (x),
(x)
c(x) = limx(x)
c(x),
Ac(R) .
It is easy to show that the constant multiples ofgc,are the only element
ofAc(R) satisfying the functional equationU(x + h) = exp(h)U(x) (andhence the differential equation U =U); in particular,
gc,(x+ h) s ehgc,(x), h =R .
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16 Asymptotic Behavior of Generalized Functions
In addition, there is an infinite number of elements ofAc(R) having S-asymptotics in the cone = R+related to c(h) with limit of the form
Cgc,(x), C R. In fact, consider (m)
, the derivatives of the Diracdelta concentrated at the origin. We have that
(m)(x + h) sc(h)mgc,(x), h =R+ ,
since
limh
(m)(x+ h), (x)
c(h) = lim
h(1)m
(m)(h)c(h)
=m gc,(x), (x) .
1.3 Equivalent definitions of the S-asymptotics inF
Theorem 1.1. LetT F and letint =. The following assertions areequivalent:
a) w.limh,h
T(x + h)
c(h)
=U(x) =Mexp(x) in
F, M
= 0 . (1.7)
b) For a-sequencemm (cf. 0.6) there exists a sequenceMmm inR, such thatMmM= 0, m , and
w.limh,h
(T m)(x + h)c(h)
=Mmexp(x), inF, uniformly inm N .(1.8)
c) For a-sequencemm, (cf. 0.6),lim
h,h
(T m)(h)c(h)
=pm, mN , (1.9)
wherepm= 0 for somem, and for every F,
suph,h0
(T )(h)
c(h)
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1. S-asymptotics inFg 17
Proof. a) b).Letmmbe a -sequence. For any F,{m ;m N} is a compact set in F.We have by the properties of the convolution(cf. 0.6) and the BanachSteinhaus theorem
limh,h
(T m)(x+ h)
c(h) , (x)
= lim
h,h
(T m)(x), (x h)c(h)
= limh,h
T, (m )( h)c(h)
= limh,h
T(x + h)
c(h) , (m )(x)
=
M ex,(m
)(x)=
M
mex, (x)
, (1.11)
uniformly in m. Now (1.11) implies (1.8) and b).
b) a).Let Fand
am,h=
(T m)(x + h)
c(h) , (x)
=(T (m ))(h)
c(h) , mN, h .
We have am,ham, h ,h ,uniformly for mN,where
am=Mmexp( x), (x), m N,ama=Mexp( x), (x), m .
Also am,hah, m ,where
ah=
T(x + h)
c(h) , (x)
, h .
This implies ah
a, h,
h
,what is in fact a).
a)c).From (1.7) it follows that (T )(h)/c(h) converges for every F,when h, h .Hence, (T )(h)/c(h) is bounded, h,h 0; (1.9) follows directly from (1.7).
c) a).First, we shall prove that the setG={m(+ x), m N, xRn}is dense inF.Suppose that T F and that
T, m( + x)= 0, m N, x Rn .It follows that (Tm)(x) = 0, m N, x Rn.Then, for any F,T m, = 0, mN,and consequently
T, = limm
T, m = limm
T m, = 0 .
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18 Asymptotic Behavior of Generalized Functions
This implies that T= 0 and hence, the setGis dense inDby the Hahn-Banach theorem. Thus, by (1.10) and the BanachSteinhaus theorem, c)
implies a).a) d).Note that (1.1) implies the strong convergence ofT( +h)/c(h)
to UinD andD respectively. Since the convolution inD andD ishypocontinuous, it follows the following equality in the sense of convergence
inE(Rn)
limh,h
T( + h)
c(h)
=
lim
h,h
T( + h)c(h)
=U
in both cases ( D,respectively D
). The paper which can be consulted for related results is [128].
1.4 Basic properties of the S-asymptotics
Theorem 1.2. LetT Fg.a) IfT(x+h)
s
c(h)U(x), h
, then for everyk
Nn0 , T
(k)(x+h) s
c(h)U(k)(x), h .b) Assume additionally thatFg is a Montel space. Let g M() (set
of multipliers ofFg (cf. 0.6)); let c, c1 be positive functions. If for every Fg, (g(x+h)/c1(h))(x) converges to G(x)(x) inFg when h,h and ifT(x +h) sc(h)U(x), h, theng(x +h)T(x +h) sc1(h)c(h)G(x)U(x), h .
c) IfT
F0 andsuppT is compact, then T(x+h)
s
c(h)
0, h
,
for every positive functionc.
d) Suppose thatFg is a Montel spaces in which the convolution withcompactly supported elements is well defined and hypocontinuous (cf. 0.6).
LetS Fg ,suppS being compact. IfT(x+h) s c(h)U(x), h, then(S T)(x + h) sc(h)(S U)(x), h.
e) LetT F = D andT(x+h) s c(h) U(x), h , inD. Assumethat (M.2) holds. LetP(D)be an ultradifferential operator of class.Then
(P(D)T) (x + h) sc(h) (P(D)U)(x), h , inD .
f ) Let T F = Q(Dn). Let P(D) be a local operator and let T(x+h)
s c(h)U(x), h, h inQ(Dn), then(P(D)T)(x+h) sc(h) (P(D)U)(x), h ,h inQ(Dn), as well.
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1. S-asymptotics inFg 19
Proof.a) The assertion is a consequence of the definition of the deriva-
tive of a generalized function. Namely,
limh,h
T(k)(x + h)
c(h) , (x)
= lim
h,h
T(x + h)
c(h) , (1)k(k)(x)
= (1)kU(x), (k)(x)=U(k), , Fg.
b) Since Fgis a Montel space, we have that limh,hT(x + h)
c(h) =U(x)inFgwith respect to strong topology. As (g(x+h)/c1(h)G(x))(x), h, h 0,is a bounded set inFg,it follows:
limh,h
g(x + h)T(x + h)/(c1(h)c(h)), (x)
= limh,h
T(x + h)/c(h),
g(x + h)
c1(h) G(x)
(x)
+ lim
h,hT(x + h)/c(h), G(x)(x)
=
U(x), lim
h,h
g(x + h)
c1(h) G(x)
(x)
+ U(x), G(x)(x)
= 0 + U(x)G(x), (x)= UG,, Fg.
c) For each F0there exists r >0 such that suppB(0, r) ={x Rn; x< r}.The support of T(x+h) is (suppT h).Thus, byour assumption, there exists rsuch that for allh,h> rthe set(suppTh)B(0, r) is empty and consequentlyT(x+h), (x)= 0, h, h r .
d) By definition of the convolution
(S T)(x + h), (x)= (S T)(x), (x h)= St Ty, (t)(t+ y h)= St Ty( + h), (t)(t+ y) .
(1.12)
Hence, (S T)(x + h) = (S T( + h))(x).
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20 Asymptotic Behavior of Generalized Functions
Since in a Montel space the weak and the strong convergence are equiv-
alent, we have by (1.12)
limh,h
(ST)(x+h)/c(h) = limh,h
S T(+h)
c(h)
(x) = (SU)(x) .
For the proof of e) and f), we note that an ultradifferential operator
P(D) maps continuouslyD intoD and a local operator maps continu-ouslyQ(Dn) intoQ(Dn).
Remark.From assertion a) of Theorem 1.2, a natural question arises
for spacesF: The limit Ucan be a constant generalized function, henceU = 0.Is there a positive functioncsuch thatT has S-asymptotics related
to thisc, but with a limit different from zero?
In general the answer is negative as shown by the following example: Let
Tbe defined byx2+ sin(expx2), x R.Then,T(x + h) sh2 1, h R+.But T(x) = 2x(1 + exp(x2) cos(expx2)).The same situation is obtained
with the distribution f(x) =x2 + x sin x.
We can now formulate an open problem: Suppose that T(x+
h) s c(h)U(x), h. If U(x) is a constant generalized function, then
the problem is to find some additional conditions onT which guarantee the
existence of a functioncsuch thatT has the S-asymptotics in related to
c.
More generally, let S F and T= (/xk)S. If T(x+ h) sc(h)U(x), h . The question is what we can say about the S-asymptoticsofS.
Recall the well known result: Let hbe a real-valued function which
has the first derivative h(x) = 0, x x0 and h(x) , x .If a function F has its first derivative on(x0, ) such that there existslim
xF(x)/h(x) =A, then there existslim
xF(x)/h(x) =A, as well.
We know that an opposite assertion does not hold, and this is at the
basis of the open problem quoted above.
We give a theorem to illustrate the relation between the S-asymptoticsof a distribution and the S-asymptotics of its primitive. We refer to [114]
for the proof.
Theorem 1.3. 1) Letf, g D(R) and for somem N, g(m) =f .
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1. S-asymptotics inFg 21
a) If f(x+h) s hL(h) 1, h R+, where >1, then g(x+h) s
h+mL(h)
1, h
R+.
b) Iff(x + h) s exp(h)L(exp h)exp(x), hR+, R, andx0
exp(h)L(exph)dh , whenx , then
g(x+h) s h
0
hm10
. . .
h10
exp(t)L(exp t)dtdh1 . . . d hm1
exp(x), h R+ .2) Let0
D(R) such that 0(t)dt= 1. If
limh
g(i)(x + h)
exp(h)L(exph), 0(x)
=iexp(x), 0(x), i= 0, 1, . . . , m 1
and
f(x + h) sexp(h)L(exph)m exp(x), h R+,
then
g(x + h) s
exp(h)L(exp h)exp(x), h
R+.
3) Suppose that T D, ={x Rn; x= (0, . . . , xk, 0, . . . , 0)} andT= (/xk)S. IfT(x+h)
sc(h)U(x), h andc(h)is locally integrableinhk such that
c1(hk) =
hkh0k
c(v)dvk as hk , h0k 0,
thenS(x + h) s c1(h)U(x), h .4) Suppose thatS D and that for anm {1, 2, . . . , n},
(DtmS)(x + h) sc(h) U(x), h .
LetV D, DtmV=U and0 D(R),R
0()d= 1. Let
limh,h
S(x + h)/c(h), 0(xm)m(x)= V, 0m ,
wherex= (x1, . . . , xm1, xm+1, . . . , xn) and
m(x) =
R
(x1, . . . , xm, . . . , xn)dxm, D .
ThenS(x + h) sc(h)V(x), h.
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22 Asymptotic Behavior of Generalized Functions
The following theorem asserts that the S-asymptotics is a local property
if the elements ofFgare compactly supported.
Theorem 1.4. LetT1, T2 F0.Let the open set Rn have the followingproperty: for everyr >0 there exists ar >0 such that the ballB(0, r) =
{x Rn; x< r} is in{h;h, h r}. If T1= T2 on andT1(x + h)
sc(h)U(x), h , thenT2(x + h) s c(h)U(x), h , as well.
Proof.Let F0with supp B(0, r).We shall prove that
limh,hT1(x + h) T2(x + h)c(h) , (x) = 0 . (1.13)The complement of the set supp(T1(x+h) T2(x+h)) contains the set{ h, h}.By our supposition the number ris fixed in such a waythat the sets{h;h, h r}contain B(0, r) and consequentlysupp.Since T1has the S-asymptotics related tocand with the limit U,
(1.13) implies
limh,h
T2(x + h), (x)
c(h) = lim
h,h T1(x + h), (x)
c(h) =
U,
.
Remark.The open set ,by its property, has to contain a set 0{xRn; x> R}where 0 is an open acute cone such that 0 andRis apositive number.
1.5 S-asymptotic behavior of some special classes of
generalized functions
1.5.1 Examples with regular distributions
1.exp(a (x + h)) s exp(a h) exp(a x), h Rn.
2.exp((x +h)2 + (x+ h))
sexph exp
x +1
2
, h R+.
3. Let w Sn1
={x Rn
; x= 1}, p= (p1, . . . , pn) Rn
+and ={qw; q R+}.Denote, J={k {1, . . . , n};wk= 0}and =iJ
pi.
Then
(x + h)p s q
iJ
wpii 1, h ,
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1. S-asymptotics inFg 23
and
(1 + x + h)p s
q iJ wpii 1, h ,
((x+ h)p = (x1+ h1)p1 . . . (xn+ hn)
pn) .
4. For a slowly varying functionL(t), t >0 we haveL(t + h)
sL(h) 1, hR+ .Namely,
limh
L(t + h)/L(h), (t)= limh
rr
(t)L(t+ h)/L(h)dt
= limq
erer
(log y)L(log(yq))/L(log q)dy
y=
R
(t)dt, D(R) .
We used above that L(log y) is also a slowly varying function (cf. 0.3)and thatL(log(uh))/L(logh) converges uniformly to 1 as h ifustaysin compact intervals [1, 2],0< 1 < 2 0 (cf. Theorem 1.2 c)).
For >0,
f(x + h) s(1/())h1 1, h R+ .
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24 Asymptotic Behavior of Generalized Functions
In the case 0, hh0.Then
limh
f(x)/h1, (x h) = limh
a
1
()
x
h
1(x h)dx
= limh
R
1
()
u + h
h
1 (u)du=
R
1
()(u)du .
Hence,f(x + h) sh1 1
(), h R+,
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1. S-asymptotics inFg 25
Property (*) is not equivalent to the existence of the S-asymptotics.
The next example illustrates that this condition is not necessary.
Assume that S C(R)L1(R) but not being bounded on R.Thisfunction has S-asymptotics equal to zero related to c= 1 (see 5.). For T
we take 1 + S(x).Then, T(x + h) s 1 1, hR+and
limh
[(1 + S) m](h) = limh
1 + S(x + h),m(x)=1, m(x)=pm .
This limit is not uniform in m Nbecause limm
[(1+S) m](h) does notnecessarily exist.
9.Every distribution in DLp , 1 p < has S-asymptotic behavior relatedtoc= 1,and =Rn,with limit U= 0.Let us show this.
By Theorem XXV, Chapter VI in [146] it follows that (T ) Lp(Rn)for every D.Every derivative of (T ),
hk(T)(h) = (T
xk)(h), h Rn,is also in Lp(Rn).Hence (T) DLp.We know
that every element of
DLp,1
p 0, c(h) 0, h, such that
|T(x + h)/c(h), (x)|M, h> D ,where andM are positive constants depending on.
The answer to this question is also negative (cf. [153]).
10.LetT Kp.Then there exists k0 N0such that Thas S-asymptoticbehavior with limit U= 0 related toc(h) exp(k0hp),wherec(h) tends toinfinity ash .
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26 Asymptotic Behavior of Generalized Functions
First, we prove that there exists a positive integerk,such that the set
{T(+ h) exp(khp), hRn}is bounded inD.We start by giving a bound for the seminorms k(( h)), Kp:
k[( h)] = supxRn,|a|k
exp(kxp)|Da(x h)|
= supxRn,|a|k
exp(kx + hp)|Da(x)|
exp(2pk
h
p) sup
xRn,|a|2pk
exp(2pk
x
p)
|Da(x)
|exp(2pkhp)2pk().
By assumption, Tis a continuous linear functional onKp.Note, thesequence of normskkis increasing. Thus, there exist >0 and k0 N0such that
|T, |1 for Kp, k0() .This inequality holds for all k k0.Hence
|T, |1k(), kk0 for every Kp.
We know thatD Kpand that the inclusion is continuous. Let D,Then
|exp(2pkhp)T(x + h), (x)|=|T(x), exp(2pkhp)(x h)|
1 exp(2pkhp)k[(x h)] 12pk(), k > k0.
We can choose k02pk.The set{exp(k0hp)T(x+h);h Rn}isbounded inD (and weakly bounded inD,as well).
Now, for every
D:
limh
exp(k0hp)T(x + h)/c(h), (x)
= limh
1
c(h)exp(k0hp)T(x + h), (x)= 0.
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1. S-asymptotics inFg 27
1.5.3 S-asymptotics of ultradistributions and Fourier hyperfunc-
tions Comparisons with the S-asymptotics of distributions
11.If a distribution has S-asymptotics inD (see Definition 1.1), it has thesame S-asymptotics inD,as well. But the opposite is not true. This isillustrated by the following example:
T(x) = 1 +
n=1
(n)(x n)/Mn, x R,
has S-asymptotics inD(Mp)(R),but it does not have S-asymptotics inD(R).We will show this. Let D(Mp)(R).Then,
n=1
(n)(x+hn)/Mn, (x)=
n=1
(1)n(n)(nh)/Mn0, h .
This is a consequence of the property
supxR
|(n)(x)|/knMn0, n for every k >0 .
Suppose that there exists a function csuch that for every D(R)
n=1
(1)n(n)(n h)/Mnc(h) ,
converges, as h .Taking h= nthis implies that (n)(0)/Mnc(n)converges to zero, as n ,for every D(R).However, such a func-tion cdoes not exist (Borels theorem). Namely, for every given sequence
Mnc(n), n
N,there exists
C0(R) such that
(n)(0)/Mnc(n)
,
as n .This example is the motivation for the following assertion.
12. Let T D(R) be such that limh
T(x+h)/c(h) exists inD(R).Assume that for some s Nand D(R) with the property
R
(t)dt= 1,
R
tj (t)dt= 0, j= 1, . . . , s ,
the following limit
limh
T(t + h)
c(h) ,
ts
ps+1
t
p
, p (0, 1],
exists uniformly in p.Then, limh
T(x + h)/c(h) exists inD(R) as well.
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28 Asymptotic Behavior of Generalized Functions
Proof.Let D(R), h h0 >0.The function Fh(t) =T(x+h+t), (x), t R,is smooth and by the Taylor formula, we have:
Fh(t) =Fh(0) + tddt
Fh(0) + + ts1(s 1)!
ds1dts1
Fh(0)
+ ts
(s 1)!
10
(1p)s1F(s)h(pt)dp, s 1,0< p 1 .
This implies
T(x + h), (x)=T(x + h + t), (x) T(x + h), (x)
(1)s1ts1(s 1)! T(x + h),
(s1)(x)
(1)sts
(s 1)!
10
(1 p)s1T(x + h +pt), (s)(x)dp,
h h0, tR .Multiplying both sides of the last equality by(t),integrating with respect
tot, and using Fubinnis theorem, we obtain
T(x + h), (x)=T(x + h + t), (t), (x) (1)s
(s 1)!1
0
(1p)s1T(x + h +pt), ts(t), (s)(x)dp .
Set
G(x,h,p) =T(x + h+pt), ts
(t), x R, h h0, p [0, 1] .We have G(x,h, 0) = 0, x R, hh0and
limh
G(x,h,p)/c(h) =Ceaxeapt, ts(t), x R, p (0, 1] ,for some a R(cf. Proposition 1.2 a)), where the limit is uniform in
p(0, 1] and xsupp. The limit function is continuous in x Randp[0, 1],because ofeapt, ts(t) 0,as p0.Because of that, weobtain
limh
1
c(h) T(x + h +pt), ts
(t), (s)
(x)
=
limh
1
c(h)T(x + h+pt), ts(t), (s)(x)
=Ceax+apt, ts(t), (s)(x)=Ceax, (s)(x)eapt, ts(t) .
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1. S-asymptotics inFg 29
This implies
limh1c(h)T(x + h), (x) = limh T(x + h+ t)c(h) , (t), (x) (1)
s
(s 1)!
10
(1p)s1eapt, ts(t)dpCeax, (s)(x) ,
which proves the assertion.
Such an assertion can be proved in the multi-dimensional case by ad-
justing the previous argument. Butan open problemis to find necessaryand sufficient conditions for a distribution T, which has S-asymptotics in
D, to have the S-asymptotics in D as well.13.We shall construct an ultradistribution out of the space of Schwartz
distributions and having S-asymptotics.
Assume (M.2) holds. LetP(D) be an ultradifferential operator of class
of infinite order (a= 0 for infinitely ) (see 0.5.2). Then, P(D)is anelement of
D which is not a distribution and which has S-asymptotics in
D equal to zero related to any c.If T D and T(x+h) s11, h inD,the ultradistribution
T+ P(D)is not a distribution, but (T+ P(D))(x+ h) s1 1, h in
D :lim
h,h((T(x + h), (x) + P(D)(x+ h), (x))
= 1, + limh,h(x + h), (1)mam(m)(x)= 1, , D .14. Let P(D) be a local operator
||0 b
D, b = 0, ||0 (see0.5.3.). The Fourier hyperfunction f= 1 + P(D)has S-asymptotics
related to c= 1 in any cone and with limit U= 1, but fis not a
distribution. For the S-asymptotics offit is enough to prove that
limh,h
P(D)(x+ h), (x)= 0, P .
SinceP(D) mapsPintoP,P(D)(x+ h), (x)=(x + h), P(D)(x)=(h) ,
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30 Asymptotic Behavior of Generalized Functions
where=P(D).By the property of the elements ofP(see0.5.3), wehave lim
h,h(h) = 0,for every cone .
A hyperfunction gsupported by the origin is uniquely expressible as
g=P(D),whereP(D) is a local operator. In such a way, the above
proof implies that every Fourier hyperfunction with support at{0}hasS-asymptotics with limit equal zero.
Since P(D)=||0
bDis a distribution if and only ifb= 0 for a
finite number of,we note that 1+P(D)is not a distribution, but it has
the S-asymptotics related to c= 1.We can also find coefficients bof a local operator P(D) such that
f= 1+P(D)is not defined by an ultradistribution belonging to the Gevrey
classD(s) orD{s} , s >1 (see [81]). For the sake of simplicity, we shallconsider the one-dimensional case. Choose P(D) such that the coefficients
of P(D) are: bn= (n! )(1+cn), n N,where cn= (log logn)1.With
these coefficients, P(D) is a local operator. Namely,
limn
nbnn! = limn(n! ) 1
n log logn = 0 .
Also, any ultradistribution in the Gevrey class s >1,supported by{0},is of the form
J(D)=
n=0
anDn,necessarily with|an|C kn/(n! )s
for some constantskand C(Beurlings type) or for anyk >0 with a con-
stant C(Roumieus type). But the coefficients bn= (n! )(1+cn) do not
satisfy these conditions, therefore P(D)cannot represent an ultradistri-
bution. Namely, since cn0 when n ,for any s >1,there exists n0such that 1 + cn< s, n n0.Thus,
(n! )(1+cn) > Ckn/(n! )s, nn0, k >0 .Consequently, P(D)does not represent an ultradistribution of Gevrey
type.
On the other hand, if we suppose that g= P(D)is an ultradistribu-tion with support{0}in the Gevrey class s >1,then, we would have anultradifferential operatorJ1(D) such that
g=J1(D)=
n=0
enDn,|en|C kn/(n! )s .
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1. S-asymptotics inFg 31
But in this case J1(D) is also a local operator and J1(D)= P(D).Thiscontradicts the fact that a hyperfunction with support at{0}is given by aunique local operator.
1.6 S-asymptotics and the asymptotics of a function
We suppose in this subsection that elements ofFare compactly supportedand, as usual, that the topology inFis stronger than the topology inE.Recall, we use the notationF0in this case.
Every locally integrable function fdefines an element ofF0(regulargeneralized functions).
We shall compare the asymptotic behavior of a locally integrable func-
tionfand the S-asymptotic behavior of the generalized function generated
by it.
A functionfhas asymptotics at infinity if there exists a positive function
c such thatlimx
f(x)/c(x) =A = 0, (in shortf(x) Ac(x), x ).1.The following example points out that a continuous andL1-integrable
function can have S-asymptotics as a distribution without having an ordi-
nary asymptotics. Suppose thatg L1(R) C(R) has the property thatg(n) =n, nNand that it is equal to zero outside suitable small intervalsIn n, n N.Denote byf(t) =et
t0
g(x)dx, t R.It is easy to see that
f(t + h) seh et
0
g(x)dx, hR+ .
By Theorem 1.2 a)f(t) has S-asymptotics related to eh and with the same
limit. But, in view of the properties ofg, f(t) =f(t) +etg(t) has not the
same asymptotics (in the ordinary sense). Moreover, gcan be chosen so
thatf has no asymptotics at all.
2.The following example shows that a function fcan have asymptoticbehavior without having S-asymptotics with limitUdifferent from zero. An
example is xexp(x2), x R.Suppose that exp(x2) has S-asymptoticsrelated to a c(h) >0, h R+with a limit Udifferent from zero. ByProposition 1.2 c), Uhas the form U(x) = Cexp(ax), C >0.Then, for
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32 Asymptotic Behavior of Generalized Functions
every F0such that >0 we havelim
h
1
c(h) exp[(x + h + h0)2](x)dx=eah0Ceax, (x) .Therefore,
eah0U, = exp(h20) limh
1
c(h)
e(x+h)
2
e2h0(x+h)(x)dx
exp(h20)U, , for every h0>0 .But this inequality is absurd. Consequently, exp(x2) cannot have such an
S-asymptotic behavior.
One can prove a more general assertion.
Proposition 1.3. LetfL1loc(R) F0(R)have one of the four propertiesfor >1, >0, x x0, h >0, M >0 andN >0 :
a) f(x + h) Mexp(h)f(x)0,a)f(x + h) Mexp(h)f(x) 0,b)0f(x + h) Nexp(h)f(x),b)0 f(x + h) Nexp(h)f(x).
Thenfcannot have S-asymptotics with limitU= 0, but the functionf canhave asymptotics.
For the proof see ([135], p. 89).
It is easy to show that for some classes of real functions fon Rthe
asymptotic behavior at infinity implies the S-asymptotics.
Proposition 1.4. a) Let c be a positive function and let T L1loc(Rn).Suppose that there exist locally integrable functionsU(x)andV(x), x Rn,such that for every compact setKRn
|T(x + h)/c(h)| V(x), x K,h> rK,lim
h,hT(x +h)/c(h) =U(x), x K .
Then, T(x + h) s
c(h)U(x), h
in
F0.
b) LetT L1loc(R) have the ordinary asymptotic behaviorT(x) exp(x)L(exp x), x , R ,
whereL is a slowly varying function. Then,
T(x + h) sexp(h)L(exph) exp(x), h R+, inF0(R) .
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1. S-asymptotics inFg 33
Proof.a) For every F0lim
h,hT(x + h)c(h) , (x) = limh,h Rn
T(x + h)
c(h) (x)dx . (1.14)
Since supp K Rn and Thas all the listed properties, Lebesguestheorem implies the result.
b) It is enough to use in (1.14) that L(yt)/L(t)1, t uniformlyin y,when ystays in compact interval contained inR+.
A more general result is the following one ([135], pp. 8990).
Proposition 1.5. Letbe a cone and letRn be an open set such thatfor everyr >0 there exists ar such thatB(0, r) { h; h,hr}. Suppose that G L1loc() and it has the following properties: Thereexist locally integrable functionsU andV inRn such that for everyr >0
we have
|G(x + h)/c(h)|V(x), x B(0, r), h ,h r;lim
h,h
G(x + h)/c(h) =U(x), x
B(0, r).
IfG0 F0 coincides withG on, thenG0(x + h)
sc(h)U(x), h .
Proof.By Theorem 1.4, it is enough to proof that
limh,h
G(x + h)
c(h) (x)dx=
Rn
U(x)(x)dx;
but as in the proof of Proposition 1.4 a), the exchange of the limit and theintegral sign is justified by our assumptions and Lebesgues theorem.
The following proposition gives a sufficient condition under which the
S-asymptotics of f L1loc(R), inD(R), implies the ordinary asymptoticbehavior off.
Proposition 1.6. Let f L1loc(R), c(h) = hL(h), where > 1and L be a slowly varying function. If for some m
N, xmf(x), x >
0, is monotonous and f(x+ h) s c(h)1, h R+,, inD(R), thenlim
hf(h)/c(h) = 1. If we suppose thatLis monotonous, then we can omit
the hypothesis >1.
For the proof see [115].
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34 Asymptotic Behavior of Generalized Functions
1.7 Characterization of the support of T F0
We suppose in this subsection that the topology inF0is defined in sucha way that a sequence{n}inF0converges if and only if there exists acompact set KRn and(k)mconverges to (k) uniformly on Kfor everyk Nn0asm .
We already proved in Theorem 1.4 a relation between the support of a
distribution and its S-asymptotics. Now, we shall complete this result.
We need a property of the S-asymptotics given in the next lemma.
Lemma 1.1. Let be a cone and let be a convex cone (it is partially
ordered). A necessary and sufficient condition that for every c(h) >0,
h a) w.lim
h,hT(x + h)/c(h) = 0 inF0,
b) w.limh,h
T(x + h)/c(h) = 0 inF0,
is that for every F0 the following holds:In case a): There exists()>0 such that
T(x + h), (x)= 0, h (), h . (1.15)
In case b): There existsh such that
T(x +h), (x)
= 0, h
h, h
. (1.16)
Proof.We have to prove only that the condition is necessary in both
cases. It is obvious that the condition is sufficient. Let us suppose the
opposite, that is, the condition is not necessary. Then, we could find a
sequenceammin such that in case a)am in ,and in case b)am in,(m ) and that
T(x + am), (x)
=m
= 0, m
N .
Let find csuch that c(h) = mfor h= am, m N.Clearly, for sucha c(h) the function h T(x+h)/c(h), (x)cannot converge to zero ash , h,or h , h.This contradicts our supposition thatthe S-asymptotics equals zero in both cases.
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1. S-asymptotics inFg 35
Theorem 1.5. Letbe a cone and letT F0.A necessary and sufficientcondition that for everyr >0 there existsr such that the sets
suppT B(h, r), h , h r are emptyis thatT(x + h)
sc(h) 0, h for every positive functionc on.
Proof.Theorem 1.2 c) and Theorem 1.4 assert that the condition in
Theorem 1.5 is necessary. We have to prove only that this condition is also
sufficient.
Let us suppose that
w.limh,h
T(x + h)/c(h) = 0 inF0 .
Let F0.By Lemma 1.1 a), we know that there exists 0() =inf(),where () is such that (1.15) holds. We shall prove that the
set{0(); F0,supp K}is bounded for every compact set KRn.Let us suppose the opposite. Then we could find a sequencekkinF0, suppkK, kN,and a sequencehkkin , hk such that
T(t + hk), p(t)=Ak,p= ak= 0, p=k0, p < k .
We give the construction of sequences {k}and {hk}.Let k F0, suppkK, kN,be such that0(k)kis a strictly increasing sequence whichtends to infinity. Then, there existhkkin andk >0, kN,such that0(k1) + k hk0(k) k.Now, we shall construct the sequencep(t)pinF0, supppK, pN,for which we have
T(t + hk), p(t)=0, p=kak, p=k .
()
Put
p(t) =p(t) p11(t) pp1p1(t), p >1, t Rn .The sequencepi iwill be determined in such a way thatp(t) satisfies (*)the sought property, p
N.
It is easy to see thatT(t+hk), k(t)= akandT(t+hk), p(t)=0, k > p.For a fixed pandk < pwe can find pi , i= 1, . . . , p 1,such thatfork= 1, . . . , p 1,
0 = T(t + hk), p(t)=Ak,p p1Ak,1 pp1Ak,p1 .
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36 Asymptotic Behavior of Generalized Functions
Hence
p1Ak,1+
+ pp1Ak,p1=Ak,p, k= 1, . . . , p
1, p >1 .
SinceAk,k= 0 for every k,the system always has a solution.We introduce a sequence of numbersbkk,
bk= sup{2k|(i)k(t)|; i k}, k N .Then, the function
(t) =
p=1p(t)/bp, t Rn, is inF0 and supp K .
Thus, we have
T(t + hk), (t)=
p=1
T(t + hk), p(t)/bp =ak/bk .
Now, if we choosec(h) such that c(hk) =ak/bk, k N,then
T(t + h), (t)
c(h) does not converge to zero whenh , h .
This proves that for every compact setKthere exists a0(K) such that
T(t + h), (t)= 0, h 0(K), h , F0,supp K.
It follows thatT(t+h) = 0 overB(0, r), h(r), h andT(t) = 0over B(h, r), h (r), h .
Remark.The condition of Theorem 1.5 implies that the support of
Thas the following property: The distance from supp Tto a point h, d(suppT, h),tends to infinity whenh , h.
The next proposition shows that if in Definition 1.1 we take the limit
(1.3) instead of the limit (1.2), then a more precise result is obtained.
Theorem 1.6. Let T F0 and be an acute, open and convex cone(partially ordered, see Remark 1 after Definition 1.1). A necessary and
sufficient condition for
suppT
CRn(a+) for somea (1.17)
is that
limh,h
T(x + h)/c(h) = 0 inF0 for everyc(h) (1.18)
(CRnA=Rn \ A).
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1. S-asymptotics inFg 37
Proof.If (1.17) holds, then for any ball B(0, 1),there exists a hrsuch that B(h, r)
(a+) for h
hr, h
.This implies (1.18), and by
Lemma 1.1, (1.17) follows.
Let us suppose now that (1.16) and consequently (1.18) hold, but for
anya,suppTCRn(a+).We fix such an a=a0 >0.There existsan a1(a0+) suppT.Since suppT CRn(2a1+),there exists ana2(2a1+ suppT).In such a way, we construct a sequenceakkinsuch thatak (kak1+) suppTandakka0, kN.
Since aksuppT, k N,it follows that there exists a sequencekkinF0such that suppkB(0, 1), k N,and
T(x + ak), k(x) = 0, k N.
We put now:
ck,i=T(x + ai), k(x), k , i N;
bk= supp{2k|(j)(x)|;jk, x Rn}, kN.
We have to prove that there exists a sequenceckksuch that ck1, kNand
k1
ck,i/(bkck) = 0, i N . (1.19)
First, we notice that ck,k= 0.If
k=1
ck,1/bk= 0,then we take c1 >1,
and if this series is different from 0,then we put c1= 1.Leti= 2.If
c1,2/(b1c1) +k2
ck,2/bk= 0 (= 0),we takec2 >1 (c2= 1)
such that
c1,1/(c1b1) + c2,1/(c2b2) +k3
ck,1/bk= 0 .
Leti= 3,
c1,3/(b1c1) + c2,3/(b2c2) +
k=3 ck,3/bk= 0 or = 0 .Then we take c3 >1 and c3= 1 respectively such that
c1,1/(b1c1) + c2,1/(b2c2) + c3,1/(b3c3) +
k=4
ck,1/bk= 0,
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38 Asymptotic Behavior of Generalized Functions
c1,2/(b1c1) + c2,2/(b2c2) + c3,2/(b3c3) +
k=4ck,2/bk= 0.
Continuing in this way, we construct a sequence ckkfor which (1.19)holds.
Let us put
k(x) =k(x)/(bkck), k Nand =ki
k.
From the properties of sequences
bk
kand
ck
k ,we can easily show
that
Nk=1
kinF0when N .
Relation (1.18) implies
T(x + ai), (x)=
k=1 ck,1/(bkck)= 0,i N.We obtain that (1.18) does not hold for .This completes the proof.
In Theorem 1.6 the support ofTcan be justCRn(a +).The question
is: Is it possible to obtain a similar proposition for the S-asymptotics given
by Definition 1.1?This question is analyzed in the next examples. We take
F0=Din which the S-asymptotics by the weak and strong convergenceare equivalent.
Examples.i) Let T(x, y) =m1 m(xm, y).The given series convergesinD(R2).Since for a D(R2), supp B(0, r),we have
limn
n
m=1
m(x m, y), (x, y)=
1mr
m(m, 0).
It follows that Tis a distribution onR2 (see Theorem XIII, Chapter 3
in [146]).
Let us remark that the support ofTlies on the half line {(p, 0)R2; p >0}.We can take for the cone R2+ {(, )R2; >0, >0}.It is a convex, open and acute cone in R2.We shall show that the limit
limh,h
T(u + h), (u)
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1. S-asymptotics inFg 39
does not exist. To do this, it is enough to take the limit over the half line
{(0, 0) +}, 0 >0,which belongs to .If we choose such that >0 and (0, 0) = 1,then for h= (p, 0)
T(u + h), (u)=m1
m(m p, 0) p.
Note, ifh ,then p ,as well. Consequently, the answer to theposed question is negative.
ii) The following example shows that if
limh,h
T(x + h)/c(h) = 0 inD
for every positive c(h),and every ={pw, p >0}, w,then this doesnot imply that
limh,h
T(x + h)/c(h) = 0 inD.Let us remark that both limits on a , whenh orh ,are equal.
LetTbe given byT(x, y) =
m1
m(x m, y 1m) on R2.The support
ofTlies on the curve
{(x, 1/x);x >0
}.Let =R2+, w= (cos, sin), 00}.Then, for a D(R2)T(x+h1, y+h2), (x, y)=
m1
m
mh1,1
mh2
0, h , h.
In order to show that
limh,h
T(x + h1, y+ h2), (x, y)does not exist, takehto belong to ={(x, a); x >0}for a fixeda >0,as
we did in example i).Remarks.As a consequence of Theorem 1.5 and Theorem 1.6, we have
some results concerning the convolution product (see [135], p. 102).
Let
G1 {f C; supp fCRn{h ; h f} ,G2 {f C; supp fCRn{h ;h hf}.
Corollary 1.1. For a fixedT D
, the convolutionT mapsD into G1if and only if the support ofThas the property given in Theorem 1.5.
Corollary 1.2. For a fixed T D and for a convex, open and partiallyordered conethe convolutionTmapsDintoG2 if and only ifsuppTCRn(a+) for somea.
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40 Asymptotic Behavior of Generalized Functions
1.8 Characterization of some generalized function spaces
Theorem 1.7. A necessary and sufficient condition for a distributionT tobelong to:
a)E is thatT(x + h) sc(h) 0, h Rn for every positivec.b)Oc is that T has S-asymptotic behavior related to every c(h) =
h, R+ and with limitU= 0.c)B is that T has S-asymptotic behavior related to every positive
c, c(h)
,
h
, and with limitU= 0.
Proof.a) It is a direct consequence of Theorem 1.5.
b) It is enough to apply Theorem IX Chapter VII in [146] which states:
A necessary and sufficient condition that a distribution T belongs to Ocis that for every D the function(T )(h) is continuous and of fastdescent at infinity.(see0.5.1).
c) By Theorem XXV, Chapter VI in [146] a distribution T
B if and
only ifT L(Rn) for every D.Suppose that T B.Then forevery Dand c(h) , h
limh,h
T(x + h)
c(h) , (x)
= lim
h,h
(T)(h)c(h)
= 0 .
Suppose that (T)(h)/c(h)0, h ,for every Dand for
every c, c(h) ash .We will show that (T)(h) L
(R
n
)for every D.Then, by the same theorem, it follows that T B.Let us assume the contrary, i.e., that (To)(h) is not bounded for
a o D.Then, we could find two sequenceshmmin Rn andcmmin Rsuch that|cm| as m ,hm mand (To)(hm) = cm.Now, for co(h) such that co(hm) =
|cm|, m N,the limit < T(x+h)/co(h), o(x) >would not exist, ash .This is in a contradictionwith our assumption that Thas the S-asymptotics related to every c(h)
which tends to infinity ash .
Proposition 1.7. a) If for every rapidly decreasing functionc, T has the
S-asymptotic behavior related toc1 and with limitUc(Uc= 0is included),
thenT S.
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1. S-asymptotics inFg 41
b) If for every rapidly exponentially decreasing function c (for every
k >0, c(h)exp(kh) 0, h ) a distributionThas the S-asymptoticbehavior related to c
1
with limitUc(Uc= 0 is included), thenT K1.
Proof.a) Letcbe given. There exists0such that for everyh,h 0,and for every D
|< T(x + h) c(h), (x)>| |< Uc, >|+M+ .Therefore, the set{T(x+h) c(h);h 0}is weakly bounded and thusbounded inD.By Theorem VI 4o,Chapter VII in [146] if{c(h)T(x+
h); h> 0}is bounded inD
,for every cof fast descent, thenT S
.b) The proof is similar to that of a), if we use the following theorem
proved which will be showed below (cf. Theorem 1.15):
LetT D. If for every rapidly exponentially decreasing functionr onRn the set{r(h)T(x + h);h Rn} is bounded inD, thenT K1.
1.9 Structural theorems for S-asymptotics inF
In the analysis of the S-asymptotics and its applications, it is useful to
know analytical expression for generalized functions having S-asymptotics,
especially if it is given via continuous functions and their derivatives. The
next theorems are of this kind. These types of theorems are usually referred
as structural theorems.
Our first result concerns the one-dimensional case, a refinement will be
obtained in Theorem 1.9 below.
Theorem 1.8. Let R and letL be a slowly varying function. Supposethatf D(R). Iff(x + h) sc(h) ex, hR+, then there is anm0Nsuch that for everym m0 the following holds:
(1) Let = 0 and c(h) = ehL(eh), h >0. Then there are gm,iC(1, ), i= 0, 1, . . . , m such that
f(x) =m
i=0 g(i)m,i(x), x (1, ),and
gm,i(x)Cixm exp(x)L(exp x), x , i= 0, . . . , m ,whereCi are suitable constants.
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42 Asymptotic Behavior of Generalized Functions
(2) Let= 0 andc(h) =hL(h), h >0. Then,
a) If >
1, then there is Fm
C(1,
) such that f= F
(m)m and
Fm(x) xm+L(x), x ;b) If 1, then there are fm,i C(1, ) and Am,i = 0, i=
0, 1, . . . , m , such that
fm,i(x)Am,ixm+iL(x), i= 0, 1, . . . , mand
f(x) =m
i=0 f(mi)m,i (x), x (1, ) .Proof.Case= 0.By the remark 1. after Proposition 1.2, there exists
a function cC(R) such that c(h)/c(x +h)exp(x), h in Rand c(i)(x + h)/c(h) i exp(x), i N0.
Since f(x+h)/c(h)exp(x) with strong convergence and the set{c(h)(x)/c(x + h);h A}is bounded inD(R),we can then use TheoremXI,Chapter III in [146] to obtain
limh
f(x + h)c(x + h)
, (x)= limh
f(x + h)c(h)
, c(h)
c(c + h)(x)
=1, (x), D(R) .Let C, (x) = 0 for x 1.We have
(x + h)f(x + h)
c(x + h) 1 in D(R) as h .
Thus,{(x+h)f(x+h)/c(x+h);h >0} is a bounded subset ofD(R).Thisimplies that this set is bounded inS(R) as well (cf. Theorem XXV, Chap-ter VI in [146]). SinceD(R) is dense inS(R),by the BanachSteinhaustheorem, we obtain
(x + h)f(x + h)
c(x + h) 1 in S(R) as h , i.e.,
limh
(f /c)(x + h)d(h)
, (x) =1, , for every S(R),where d(h) = 1, h > A.We shall now borrow two theorems about quasi-
asymptotics (see Definition 2.2) which will be shown later. Since the S-
asymptotics inS+with >1 implies the quasi-asymptotics of f /c
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1. S-asymptotics inFg 43
(see Theorem 2.47), the structural theorem for the quasi-asymptotics (see
Theorem 2.2) implies that there is m0 Nsuch that for every m > m0there is FmC(R) such that
(f /c)(x) =F(m)m (x), x R,andFm(x)xm as x .Thus, we obtain
f(x) = c(x)F(m)m (x), x (1, ).The Leibnitz formula implies
f(x) =
mi=0mi (1)i(c(i)(x)Fm(x))(mi), x (1, ).
Since
c(i)(x+ h)
c(h) iex, h , x R,
we obtain
c(i)(h)
ic(h), h
.
This implies the result when = 0 and c(h) =ehL(eh), h >0.In case= 0 andc(h) =hL(h), h >0, > 1,the S-asymptotics of
frelated to c(h) =hL(h), h >0, >1,implies the quasi-asymptoticsoffrelated to the same c(h) (see Theorem 2.47). The assertion follows
now from Theorem 2.2.
If 1,then we take k >0 such that k+ >1.With as in thepreceding proof and by Theorem 1.2 b), we have
(1 + (x+ h)2)k/2(x+ h)f(x + h) s hk+L(h) 1, h R+.
By the same arguments as in the preceding proof, we have that there is
m0 Nsuch that for every m > m0there is an Fm C(R), suppFm(0, )
Fm(x)x+k+mL(x), x
(1 + x2)k/2(x)f(x) =F(m)m (x), x R.Thus, for x (1, ),
f(x) =
mi=0
(1)i
m
i
1
(1 + x2)k/2
(i)Fm(x)
(mi).
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44 Asymptotic Behavior of Generalized Functions
The result now follows from the fact
1(1 + x2)k/2(i)
Cixki
, x ,whereCi= 0 are suitable constants,i= 0, . . . , m .
We can show a more general result which describes the precise structure
of a distribution having S-asymptotics.
Theorem 1.9. If T D has S-asymptotics related to the open coneandthe continuous and positive function c(h), h, then for the ball B(0, r)there exist continuous functions F
i,|i|
m, such that Fi(x+h)/c(h) con-
verges uniformly forxB(0, r) whenh, h , and the restrictionof the distributionT onB(0, r)+can be given in the formT=
|i|m
DiFi.
We need the following lemma for the proof of the theorem:
Lemma 1.2. Let T D, T(x+h) s c(h)U(x), h. Then, for anopen ball B(0, r) and a relatively compact open neighborhood of zero in
Rn
, there exists an m N0 such that for every , Dm the function
(T)(x)is continuous forx B(0, r)+.Moreover, the set of functions{(Th )(x);h}converges uniformly forx B(0, r)to(U)(x),ash , h ;Th=T(x +h)/c(h).
Proof.Suppose thatThas S-asymptotics related toc(h).Then, the set
{T(x+h)/c(h) Th;h}is weakly bounded inD and consequently,bounded inD.A necessary and sufficient condition that a setB D isbounded inD
is: for every Dthe set of functions{T ;T B
}is bounded on every compact set K Rn (see7, Chapter VI in [146]).Moreover{T ;T B}defines a bounded set of regular distributions.
LetCl() =K(Cl() is the closure of ); Kis a compact set. For a
fixed C0 , supp K,the linear mappings (Th ) , h,are continuous mappings ofDKintoEbecause of the separate continuityof the convolution. Since the set{Th ; h}is a bounded set inD,forevery ball B(0, r) the set of mappings
{(Th
)
;h
}is the set
of equicontinuous mappings ofDKintoLB,whereB=B(0, r).Now thereexists anmN0such that the linear mappings (, )Th whichmapDK DKinto LBcan be extended toDm Dmin such a way that(, )Th , h,are equicontinuous mappings ofDm DmintoLB(see for example the proof of Theorem XXII, Chapter VI in [146]).
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1. S-asymptotics inFg 45
We proved that for every , Dmand every h the functionsx (Th )(x) are continuous in x B(0, r).From the relation(Th )(x) = (T )(x+h)/c(h) and from the properties ofcitfollows thaty(T )(y) is a continuous function for y B(0, r) + and, Dm .
It remains to prove thatTh converges toUas h , h,in LBfor , Dm .We know thatDis a dense subset ofDm, m0.We can construct a subset AofDKto be dense inDm. The set of functionsThconverges inLBfor, A,whenh , h .Taking care ofthe equicontinuity of the mappings
Dm
DmintoL
B ,defined byTh
,
we can use the BanachSteinhaus theorem to prove thatThconvergesin LBwhen h , h .
Proof of the Theorem.We shall use (VI, 6; 23) from [146].
2k (E E T) 2k (E T) + ( T) =T , (1.20)
whereEis a solution of the iterated Laplace equation kE=; , D.We have only to choose the natural number klarge enough so that Ebelongs
toDm .Now, it is possible to take F1= E E T, F2=E TandF3= T.All of these functions are of the formFi=Ti , i, , Dm , i= 1, 2, 3.
The following holds:
Fi(x + h)/c(h) = (Fi(x)/c(h)) h= (T i i, )(x) h/c(h)= ((T(x) h)/c(h) (i i) = (Th i i)(x), x B(0, r), h .
Hence, by Lemma 1.2 it follows that Fi(x+h)/c(h) converges uniformly
forx B(0, r) whenh, h .
Consequences of Theorem 1.9
a) If the functions Fi, |i| m,have the property given in Theorem 1.9and if =Rn,then the regular distributions defined by the functionsFi/c
have the S-asymptotic behavior related to c1(h) 1.
b) If T S,then all functions Fi, |i| m,are continuous for xB(0, r) + ,and of slow growth (Fi= (1 +r2)qfi, |i| m, q R,wherer= xand fi,|i|m,are continuous and bounded functions).
c) The converse of Theorem 1.9 is also true. Therefore, it completely
characterizes those distributions having S-asymptotics.
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46 Asymptotic Behavior of Generalized Functions
Proof.a) Suppose that = Rn.Then by the properties ofFi, |i| m,the functions Fi/care continuous andFi(x)/c(x) converge to the numbers
Ciasx .Now, for|i| mwe havelim
hFi(x + h)/c(x + h), (x)
= limh
Rn
(Fi(x + h)/c(x + h))(x)dx= Ci, , D.
b) IfT S,then there exists a q Rsuch that the set of distributions
{T(x+h)/(1 +
h2)q;h
Rn
}= Wis bounded in
D (Theorem VI,
Chapter VII in [146]). We can now repeat the first part of the proof of
Lemma 1.2 but with c(h) = (1 +h2)q, h = Rn.In this may weobtain that there exists a pN0such that for , Dpand xRn thefunction x(T)(x) is continuous and (T)(x)/(1+x2)q, x Ris bounded. It remains only to choose the numberkin (1.20) large enough
so thatE Dmax(m,p) .c) It follows directly from Theorem 1.2 a) and Proposition 1.5.
We can also characterize the structure of ultradistributions having S-asymptotics.
Theorem 1.10.([132]). Let T D . Suppose: (M.1), (M.2) and (M.3)are satisfied byMp; = 1+1,where1Rn is open set and1 Rn isconvex cone;is a subcone of1.ThenThas the S-asymptotics related to c
andif and only if for a given open and relatively compact setA(A ),there exist an ultradifferential operator P(D) of class
and continuous
functionsf1 andf2 onA + such that
limh,h
fi(x + h)/c(h), i= 1, 2,
exist uniformly inx A, i= 1, 2, andT=P(D)f1+ f2 onA + .
Proof.One can easily prove that the condition is sufficient.
The condition is necessary. Suppose that T(x+ h) s
c(h)
U(x), h
inD.Denote by Th= hT /c(h) and by (CB)Athe space of continuousand bounded functions onA.By Theorem 6.10 in [79],Fh: Th, h,are continuous mappings: DBr+ (CB)A.Consequently, Fhare thecontinuous mappings:DK(CB)A,where K=B r .The set {Th; h,h}, >0,is bounded inDbecause of the S-asymptotics ofT.
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1. S-asymptotics inFg 47
Thus, for a fixed DK,the set{Fh(); h,h }is bounded in(CB)A.Since DKis a barrelled space, by the Banach theorem it follows thatthe family of functions{Fh}={Fh; h, h }is equicontinuous.Therefore, there exists Hp, p Nof the form (0.1) and >0 such thatany function in{Fh}maps the neighborhood of zero
V=
DK; sup
xK,||N0
|()(x)|/H||M||<
(1.21)
into the unit ball B(0, 1) in (CB)A.Denote byDHpMpK the completion of
DKunder the normqHpMp(see (0.5)).
Using the extension of a function through its continuity (see [12]), we
shall show that the family{Fh}can be extended onDHpMpK , keeping theuniform continuity; let us denote it by{Fh}.
Let DHpMpK and letjjbe a sequence inDKwhich converges tobeing a Cauchy sequence in the norm qHpMp .We shall prove thatTh jconverges, as j ,in (CB)Auniformly in h,h.It is enoughto prove thatTh jjis a Cauchy sequence in (CB)A.
Every neighborhood of zero Win (CB)Acontains the ball B(0, ) forsome >0.The neighborhood of zero V ,given by (1.21), satisfies ThV W,when h ,h .
Letj0 N0be such that i jVifi, jj0.Then,Th i Th j=Th (i j) W, h ,h .
This proves the existence of the family{Fh}and that limj
Th j=gh (CB)A,h .We shall prove thatgh=Th .The sequencejjconverges to also inEBr+ , as well. So, Th jconverges to Th as
j ,inD,where = Br+(see [79], p. 73). Thus, (Th )|Amust be gh.
It remains to prove that for DHpMpK , Th converges in (CB)Aash,h .Since{Fh}is an equicontinuous family of functions, forevery DHpMpK the set{Th ; h,h }is bounded in (CB)A.By the BanachSteinhaus theorem Th converges in (CB)A.
Now we will give the analytic form of T.Let DQsuch that Qisa compact subset of the interior ofKand qHpMp()
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48 Asymptotic Behavior of Generalized Functions
byf1andw Tby f2.The properties of these two functions follow by theprevious part of the proof.
We pass now to the case of Fourier hyperfunctions.
Theorem 1.11. Letf= [F] Q(Dn), F O((Dn+iI#Dn).Iffhas theS-asymptotics related to c and, then there exist an elliptic local operator
J(D) and functionsqs C(Rn), s , is the set ofn-vectors with entry{1, 1}), of infra exponential type such that:
1. qs(z) O(Dn + iIs),whereDn + iIs is an infinitesimal wedge of type
Dn
+ is0, s.2. f=J(D)
s
qs(x).
3. For everys there exists0>0 such thatqs(x + is)/c(x), s converge asx , x , for every fixed,0< < 0.
Proof.Let
f= F(x + i), F= sgnF
and
F(f) = [R]=
F(F)( i0) .
Then there exists a monotone increasing continuous positive valued function
(r), r 0,which satisfies (0) = 1, (r) , r and such that
|R()|Ckexp(||/(||),1k |Im j |1 ,
wherej= 1, . . . , nand k N(cf. [74], p. 652).By Lemma 1.2 in [73], we can choose an entire function Jof infra
exponential type on Cn which satisfies the estimate:
|J()|Cexp(||/(||)), |Im |1 .Then J2()O(Dn + i{||
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1. S-asymptotics inFg 49
Denote byg=F1(1/J2).By Theorem 8.2.6 in [75], g Q1(Dn).ByProposition 8.4.3. in [75]
f g= R
f( t)g(t)dt Q(Dn)
and
F(f g) =F(f)F(g) .By the Corollaries of Proposition 2 in [165]
f=J0(D)(g f), J0=J2
. (1.22)
We can always assume that there exists R+such thatI= (, )n.Then we denote by I=I , .
By Proposition 8.3.2 in [75] the following assertions are true:
F O(x+iI) and it decreases exponentially outside any conecontaining as a proper subcone.
F(F)
O(x
iI) and it decreases exponentially outside any cone
containing as a proper subcone.
F(F)J2 has the same cited properties asF(F).F(F)J2sO(x iIs) and it decreases exponentially outside any
cone containing and sproper subcone.
F1(F(F)J2s) O(x+ i(I Is)) and it decreases exponentiallyoutside any cone containing sas a proper subcone.
Consider now the Fourier hyperfunctionf ggiven in (1.22):f g=F1(F(f)F(g))
= 1(2)n
Rn
eizF(F)()/J2()d ,
where Iandz Rn + iI.Let be fixed. Then for and zRn + iI,we have
S,(z) = 1(2)n
Rn
eizF(F)()/J2()d;
|S,(z)| 1(2)n
Rn
exyF(F)()/J2()d .
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50 Asymptotic Behavior of Generalized Functions
One can see that S,(z), z Rn +Iare continuable to the realaxis. The corresponding functionsxS,(x) are continuous and of infraexponential type on R
n
.By Lemma 8.4.7 in [75], for x Rn
,S,(x)=S(x + i0),,
(f g)(x) =
S,(x), x Rn .(1.23)
FunctionsS,can be written in the form
S,
(z) =
1
(2)n Rn
eiz
F(
F
)(
)/J2(
)
s(
)d, z
Rn+I
.
Lets .Then functionsS,,s(z) =
1
(2)n
Rn
eizF(F)()/J2()s()d ,
zRn + iI, ,, s ,are also continuable to the real axis. The corre-sponding functions x S,,s(x) are continuous and of infra exponentialtype onR
n
.Moreover, on Rn
,S,,s(x)=S,,s(x + i0) ,
S,(x) =s
S,,s(x) .(1.24)
Let us analyze functions
Is() =J2()ess(), (Rn + i{||0.These functions are elements ofPbecause of
|Is,()|=|J2()|exp
n
i=1sii
ni=1|si(i)|
|J2()|ni=1|si(i) exp(sii)Cexp(
n
i=1|i|), ||
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1. S-asymptotics inFg 51
Denote by
qs(x) = S,,s(x)
=
F1(F(F)J2)(x + i(s)0), x Rn . (1.25)
We prove that functions qs, s,have the properties cited in Theorem1.11.
Property 1 follows from (1.24) and (1.25).
By (1.22) and (1.23) property 2 is satisfied.
It remains to prove property 3.
If f Q(Dn) and P,then f O(Dn +iI),where I isan interval containing zero. We shall use this fact and the properties of
functionsIs,,already analyzed.
For a fixed s there exists 0 >0 such that sbelongs to all in-finitesimal wedges of the formDn + i(
s)0 which appear in (1.25). For
(0, 0],we have
qs(x + is) =
1
(2)n
Rn
ei(x+is)F(F )()J2()si()d
=
1
(2)n
Rn
eixF(F )()F(s,)()d
=
((F ) s,)(x)
=
F
s,
(x) = (f s,)(x).
Now, for every fixed (0, 0],ands,
limx,x
qs(x + is)/c(x) = limx,x
(f
s,)(x)/c(x)
= limx,x
f(t+ x)/c(x), s,(t).
We cite some papers related to this problem: ([131], [132], [155],[151], [123]).
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52 Asymptotic Behavior of Generalized Functions
1.10 S-asymptotic expansions inFg
A sequencennof positive real-valued functionsn(t), t(t0, ), t0 0(defined on (0, t0), t0 >0) is said to be asymptoticif and only ifn+1(t) =o(n(t)), t (t0).The formal series
n1
un(t) is an asymptotic
expansion of the function urelated to the asymptotic sequence{p(t)}if
u(t) k
n=1
un(t) =o(k(t)), t (t0) (1.26)
for everykN.We write in this case:
u(t)
n=1
un(t)|{n(t)}, t (t0). (1.27)
Ifun(t) =cnn(t),for every nN, where cnare complex numbers, thenexpansion (1.27) is unique. Indeed, the numbers cncan be unambiguously
computed from (1.26). In this case, we omit from the notation{
n(t)}in
(1.27). A series which is an asymptotic expansion of a function fcan be
also convergent. However, the series is divergent in general; nevertheless,
several terms of it can give valuable information, and very often its good
approximation properties come actually from the fact that the series is
divergent. Sometimes, if we take more terms from the asymptotic series,
we obtain a worse approximation; consequently, the determination of the
optimal number of terms for a good approximation depends on a careful
analysis of the problem under consideration.
An asymptotic expansion does not determine only one function. The
following example illustrates this fact:
exp
1
x
k=0
xk
k! and exp
1
x
+ exp(x2)
k=0
xk
k! , x .
In many problems of applied mathematics one is led to the use of asymp-
totic series. (See [44], [10], [15], [14], [192], [69]). A clear exposition of thetheory and the use of asymptotic series of functions and distributions can
be found in ([50][56]).
We shall discuss in this section the S-asymptotic expansion of general-
ized functions belonging toFg.
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1. S-asymptotics inFg 53
1.10.1 General definitions and assertions
In this section will be a convex cone with the vertex at zero belonging toRd and () the set of all real-valued and positive functions c(h), h.We shall consider the asymptotic expansion whenh , h .
Definition 1.2.A distribution T Fghas a S-asymptotic expansion re-lated to the asymptotic sequencecn(h)n (),if for every F
T(t + h), (t)
n=1Un(t, h), (t)|{cn(h)}, h , h , (1.28)
where Un(t, h) Fg(with respect to t) for nNand h.We write inshort:
T(t + h) s
n=1
Un(t, h)|{cn(h)}, h , h . (1.29)
Remarks.1) In the special caseUn(t, h) =un(t)cn(h), un Fg, nN,we simply write
T(t + h) s
n=1
un(t)cn(h), h , h . (1.30)
In this case the given S-asymptotic expansion is unique.
2) Brychkovs general definition is inS(R) and slightly different fromours ([15], [16] and [20]); his idea reformulated inFg(R) gives the followingdefinition. Suppose thatf
Fg(R) and that the function exp(ixh),where
his a real parameter, is a multiplier inFg(R).
Definition 1.3.Suppose that f Fg(R).It is said that f(x)eixh has anasymptotic expansion related to the asymptotic sequencen(h)nif forevery Fg(R)
f(x)eixh, (x)
n=1Cn(x, h), (x)|{n(h)}, h ,
where Cn(x, h) Fg(R) (with respect to x), nN, hh0.We write inshort:
f(x)eixh
n=1
Cn(x, h)|{n(h)}, h .
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54 Asymptotic Behavior of Generalized Functions
To obtain an equivalent definition of this asymptotic expansion,
Brychkov has supposed thatFg(R) =S(R) andFg(R) =S(R).Then, byputtingg=f=F(f), cn(, t) =F(Cn(, t)) and =F(),Definition 1.3reduces to:
Definition 1.4.A distribution g S(R) has an asymptotic expansionrelated to the sequencen(h)nif for every S(R)
g(h t), (t)
n=1
cn(t, h), (t)|{n(h)}, h ,
wherecn(, h) S(R), n Nandhh0.
Definition 1.4 is, of course, a particular case of Definition 1.2 if we use
T(t) =g(t) and the cone =R.In [15] and [20] authors studied asymptotic expansions of tempered dis-
tributions given by Definition 1.3. In [16] Brychkov extended Definition 1.4
to then-dimensional case, but only on a ray{y; >0}for a fixedy Rn.We study in this section the asymptotic expansion not only inS(R)
and not only on a ray, but on a cone in Rn.The next remarks state some
motivations for such investigations.
Remarks.1) A distribution inS(R) can have an S-asymptotic expan-sion inD(R) without having the same S-asymptotic expansion inS(R).Such an example is the regular distribution fdefined by
f(t) =H(t) exp(1/(1 + t2))exp(
t), t
R ,
whereHis the Heaviside function.
It is easy to prove that forh R+
f(t+h) s
n=1
1
(n 1)! (1+(t+h)2)1n exp(th)|{ehh2(1n)}, h ,
while
tUn(t, h) = (1 + (t+ h)2
)1n
exp(t h), n N, h >0do not belong toS(R).
2) The regular distribution gdefined by
g(t) = exp(1/(1 + t2))exp(t), tR
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1. S-asymptotics inFg 55
belongs toD(R) but not toS(R).It has the S-asymptotic expansion inD(R) :
g(t+ h) s
n=1
1
(n 1)! (1 + (t+ h)2)1n exp(t+ h)|{ehh2(1n)}, h ,
where =R+.
3) We can distinguish two cases of S-asymptotic expansions. If in
Definition 1.2,
Un(t, h) =un(t)cn(h), nN ,
then the S-asymptotic expansion iscalled of second type. (See Remark
after Definition 1.2). IfUn(t, h) =un(t+h), n N,then theS-asymptoticexpansion is of first type.
The following example illustrates the difference between these two types
of S-asymptotic expansions.
Let f(x) = x2 + x , x >0 and f(x) = 0, x0.A S-asymptoticexpansion of the first type for this distribution is
f(x + h) s
n=1
1/2
n 1(x + h)2n|{h2n}, h ,
but the sequenceun(x) =
1/2
n1x2n cannot give a S-asymptotic expansion
of the second type for f.S-asymptotic expansions have similar properties as those of S-
asymptotics.
Theorem 1.12. LetT Fg and
T(t + h) s
n=1 Un(t, h)|{cn(h)},h , h .Then:
a) T(k)(t+ h) s
n=1
U(k)n(t, h)|{cn(h)},h , h .
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56 Asymptotic Behavior of Generalized Functions
b) Let the open set have the property: for everyr >0 there exists a
0 such that the closed ball B(0, r) ={xRn, x r} is in{ h, h, h 0}. IfT, T1 F
0 andT1=T over, then
T1(t + h) s
n=1
Un(t, h)|{cn(h)},h , h ,
as well.
c) Assume additionally thatFg is a Montel space and that inFg theconvolution is well defined (see 0.6) and hypocontinuous. Let S Fg,supp S being compact. Then
(S T)(t+ h) s n=1
(S Un)(t, h)|{cn(h)},h , h .
Proof.We prove only a). The proofs of b) and c) are the same as in the
proof of Theorem 1.2. We have
limh,h
T(k)(t+ h)
mn=1
U(k)n(t, h), (t)
cm(h)
= limh,h
T(t + h)
mn=1
Un(t, h), (1)|k|(k)(t)
cm(h) = 0 .
A relation between the asymptotic expansion of a locally integrable func-
tion fand its S-asymptotic expansion when seen as a regular generalized
function is provided in the following proposition (see [152]).
Proposition 1.8. Let f(t), Un(t, h) and Vn(t), t Rn, n N, h,be locally integrable functions such that for every compact setKRn the
following ordinary asymptotic expansion holds,
f(t + h)
n=1
Un(t, h)|{cn(h)},h , h, t K
and for everykN1
ck(h) f(t + h) k
n=1 Un(t, h) Vk(t), t K, h , h r(k, K) .Then forf F0, we have
f(t + h) s
n=1
Un(t, h)|{cn(h)},h , h .
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1. S-asymptotics inFg 57
Proposition 1.9. Suppose that has the nonempty interior. LetT Fand
T(t + h) s
n=1
un(t)cn(h),h , h .
Ifum= 0, m N, thenum has the form
um(t) =
mk=1
Pmk(t) exp(ak t), tRn, mN ,
where ak = (ak1 , . . . , akn) Rn and Pmk are polynomials with degrees less
thank at every ti, i= 1, . . . , n .
Proof.Definition 1.2 and the given asymptotics implies
limh,h
T(t + h)/c1(h) =u1(t) = 0 in F .
Now Proposition 1.2 implies the explicit form ofu1.
The following limit givesu2:
limh,h
T(t + h), (t) u1(t), (t)c1(h)c2(h)
=
u2, ,
F.
Note,
limh,h
(Dti a1i )T(t + h), (t)c2(h)
=(Dti a1i )u2(t), (t), F.
Two cases are possible.
a) If (Dti a1i )u2= 0, i= 1, . . . , n ,then u2(t) =C2exp(a1 t).
b) If (Dti a1
i )u2= 0 for some i,then by Proposition 1.2, (Dtia1i )u2(t) =c exp(a2 t) and u2has the formC2exp(a
1 t) + P22 (t1, . . . , tn)exp(a2 t) ,where P22is a polynomial of the degree less than 2 with respect to each
ti, i= 1, . . . , n .
In the same way, we prove the assertion for every um.
We will give an example of a function which has S-asymptotic expansion
of the first type but does not have the asymptotic expansion as a function:
Let (t) = 1, t(n 2n, n+ 2n), n N,and (t) = 0 outside ofthese intervals. Let
(x) =ex
x0
(t)dt, x R, R .
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58 Asymptotic Behavior of Generalized Functions
Sincex0
(t)dt2 as x ,we have that (x)2ex, x but(x) does not have an ordinary asymptotic behavior (see Example 5. in
1.5.1).Letjjbe a strictly decreasing sequence of positive numbers. Let
be a function, C, 1 for x >1, 0 for x
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1. S-asymptotics inFg 59
We write in short