Existence and uniqueness of $v$-asymptotic expantions and Colombeau's generalized numbers

0.26 MB·English

by  Todor D. Todorov

v
6 Existence and Uniqueness of -Asymptotic
0
0
Expansions and Colombeau’s Generalized
2

n
Numbers
a
J

0
3 Todor D. Todorov

] Mathematics Department
A
C California Polytechnic State University
h. San Luis Obispo, California 93407, USA
t
a ([email protected])
m
[

1
Abstract
v
0
Wedefineatypeofgeneralizedasymptoticseriescalledv-asymptotic.
2
7 We show that every function with moderate growth at infinity has a
1 v-asymptotic expansion. We also describe the set of v-asymptotic
0
series, where a given function with moderate growth has a unique
6
0 v-asymptotic expansion. As an application to random matrix the-
/ ory we calculate the coefficients and establish the uniqueness of the
h
t v-asymptotic expansion of an integral with a large parameter. As an-
a
m other application (with significance in the non-linear theory of gener-
: alized functions)we show that every Colombeau’s generalized number
v
hasav-asymptoticexpansion. AsimilarresultfollowsforColombeau’s
i
X
generalized functions, in particular, for all Schwartz distributions.
r
a
Mathematics Subject Classification: 30B10, 34E05, 35D05, 41A60, 40A30, 46F30.
Key words: asymptotic expansion, valuation, ultrametric space, Colombeau gen-
eralized functions, random matrix theory.
1 Introduction
Our framework is the ring of functions with moderate growth at infinity, sup-
M
plied with a pseudovaluation v : R and pseudometric
M → ∪{∞}
1
dv : 2 R (Section 2). We denote by v the ring of functions with the
M → F
non-negative pseudovaluation and by the set of functions with zero pseudoval-
v
C
uation. Here are some typical examples: r +1/xn v, r R, n = 0,1,2,...
∈ F ∈
and sinx,cosx,lnx .
v
∈ C
In Section 4 we prove the existence of a linear homomorphism (a linear oper-
ator) st, called the “pseudostandard part mapping,” from into with range
v v
F F
within 0 . The mapping st is an extension of the limit lim , considered
v x
C ∪{ } →∞
as a fubnctional (part (iii) of Theorem 6).
In Section 5 we define the cboncept of v-asymptotic series; these are series of
the form ∞n=0 ϕxnr(nx), where (rn) is a strictly increasing (bounded or unbounded)
sequencein R andtheset ϕn(x) n = 0,1,... isalinearlyv-independentsubset
P { | }
of (Section3). Inthatsense,thefunctionsin playtheroleof“ourgeneralized
v v
C C
numbers.” There are many examples in this section: some series are both v-
asymptotic and asymptotic (in the usual sense), some series are v-asymptotic, but
not asymptotic and others are asymptotic, but not v-asymptotic.
The main result is in Theorem 8 (Section 7): Every function with moderate
growth at infinity has a v-asymptotic expansion. We also describe the sets of
v-asymptotic series, where every function in has a unique v-asymptotic ex-
M
pansion. We shall try to explain the idea of the proof by modifying a familiar
theorem in asymptotic analysis: Let f(x) be a function, (r ) be a strictly in-
n
creasing sequence in R and (cn) be a sequence in C. The asymptotic expansion
f(x)∼ ∞n=0 xcrnn, x→ ∞, holds iff
P n 1
(1) c = lim xrn f(x) − ck ,
n x " − xrk#!
→∞ k=0
X
for n = 0,1,2,.... This result can be rephrased as follows: If all limits (for
n = 0,1,2,...) on the RHS of (1) exist, then f(x) has a unique asymptotic expan-
sion of theform ∞n=0 xcrnn. Itis clear that mostof thefunctionsin Mdonothave
asymptotic expansions of this form. Actually, there are very few general existence
P
results in asymptotic analysis (N. G. De Bruijn [1], R. Estrada and R. P. Kan-
wal [5]). This is, perhaps, the reason why many consider the asymptotic analysis
as a “kind of art” or “collection of different methods,” rather than a maturemath-
ematical theory. Our article is an attempt to improve this situation. Here is the
way we modify the above asymptotic theorem:
(a) Instead of series of the form ∞n=0 xcrnn, we consider v-asymptotic series
∞n=0 ϕxnr(nx), where v(ϕn)= 0 and v iPs the pseudovaluation mentioned earlier.
(b)Wereplacelim (consideredasafunctional)inthecounterpartof(1)by
x
P →∞
the linear operator st mentioned earlier. The advantage of st over lim is that
x
→∞
st is defined on the whole (including functions such as sinx,cosx,lnx,ln(lnx),
v
F
b b
b 2
etc.). Incontrast,lim isdefinedonlyonapropersubsetof (e.g. lim sinx
x v x
→∞ F →∞
and lim lnx do not exist).
x
→∞
(c) Instead of a (fixed) sequence (r ) (which corresponds to a choice of a fixed
n
asymptoticscale),weallowdifferentsequences(r )forthedifferentfunctionsf(x).
n
Two applications are presented in the article: At the end of Section 7 we prove
theuniquenessofthev-asymptotic expansionofanintegral withalargeparameter
withorigininrandommatrixtheory. InSection 8weshowthateveryColombeau’s
generalized number has a v-asymptotic expansion. A similar result follows for
Colombeau’s generalized functions, in particular, for all Schwartz distributions.
This result has importance in the nonlinear theory of generalized functions (J.F.
Colombeau [2]-[3]) and its applications in ordinary and partial differential equa-
tions (M. Oberguggenberger [11]).
ThisarticlehasmanyfeaturesincommonwithanotherarticlebyT.D.Todorovand
R. Wolf [13] on A. Robinson’s asymptotic numbers in the framework of the non-
standard asymptotic analysis (A. H. Lightstone and A. Robinson [9]).
In what follows N = 1,2,3,..., denotes the set of natural numbers and R
{ }
and C denote the fields of the real and complex numbers, respectively.
2 Our Framework: Functions of Moderate
Growth at Infinity
We define and study the properties of a pseudovaluation and the corresponding
pseudometric in the ring of complex valued functions with moderate growth at
infinity. Most of the results are elementary and presented without proofs.
Definition 1 (Functions of Moderate Growth) (i) We denote by G (R,C)
∞
the ring (under the usual pointwise addition and multiplication) of all complex
valued functions f defined on dom(f) R such that (n, ) dom(f) for some
⊆ ∞ ⊆
n N. We shall treat the elements of G (R,C) as germs at , i.e. two functions
∈ ∞ ∞
will be identified if they have the same values for all sufficiently large x. We define
a partial order in G (R,C) by f > 0 if f(x)> 0 for all sufficiently large x.
∞
(ii) The functions in
= f G (R,C) f(x)= O(xn), as x , for some n N
M { ∈ ∞ | → ∞ ∈ }
are called functions with moderate growth at (or, moderate functions,
∞
for short) and those in
= f G (R,C) f(x)= O(1/xn), as x , for all n N
N { ∈ ∞ | → ∞ ∈ }
3
are called null-functions.
(iii) We define a pseudovaluation v : R by
M → ∪{∞}
(2) v(f) = sup r R f(x)= O(1/xr) as x .
{ ∈ | → ∞}
The functions in the sets:
(3) = f v(f)> 0 ,
v
I { ∈ M | }
(4) = f v(f) 0 ,
v
F { ∈M | ≥ }
(5) = f v(f) = 0 ,
v
C { ∈ M | }
are called v-infinitesimal, v-finite and v-constants, respectively. The functions
in will be called v-infinitely large.
v
M\F
(iv) If S G (R,C), then the set:
⊆ ∞
(6) µ (S) = ϕ+dϕ ϕ S, dϕ
v v
{ | ∈ ∈ I }
is called the v-monad of S. In the particular case S = f , we shall write µ (f)
v
{ }
instead of the more precise µ ( f ).
v
{ }
Example 1 : xr, eix, sinx, cosx, lnx, ln(lnx) are all in (where r R).
M ∈
Also, 0 and e x are in . In contrast, ex / . Notice that is subring of
−
N ∈ M M
G (R,C) with zero divisors. In particular, the functions xr, eix, lnx, ln(lnx) are
∞
all multiplication invertible in , while sinx,cosx are not.
M
Example 2 : If r R, then v(1/xr) = r. More generally, v(P) = deg(P) for
∈ −
any polynomial P C[x], where deg(P) denotes the degree of P. We have v(c) = 0
∈
forallc C, c = 0. Inotherwords, C 0 v. Also, sinx, cosx, eix, lnx, lnnx
∈ 6 \{ } ⊂ C
and ln(lnx) are all in , because v(eix) = v(sinx) = v(cosx) = v(lnx) =
v
C
v(lnnx) = v(ln(lnx))= 0 (n Z). Finally, v(0) = v(e−x)= .
∈ ∞
The results of the next three lemmas follow immediately from the definition of
the pseudovaluation:
Lemma 1 (A Characterization of v) Let f G (R,C) and r R. Then the
∈ ∞ ∈
following are equivalent:
(i) v(f)= r.
(ii) 1 f(x) √nx for all n N.
xr√nx ≤ | | ≤ xr ∈
ϕ(x)
(iii) f(x)= for some ϕ .
xr ∈Cv
4
Lemma 2 (A Characterization of ) Let f G (R,C). The following are
N ∈ ∞
equivalent:
(i) f .
∈ N
(ii) v(f)= .
∞
(iii) limx xnf(x) = 0 for all n N.
→∞ ∈
Lemma 3 (Connection with O’s Symbols) (i) o(1) in the sense
v v
I ⊂ ⊂ F
that f implies f(x)= o(1), as x , which implies f .
v v
∈ I → ∞ ∈ F
(ii) O(1) o(1) in the sense that if f(x) = O(1) and f(x) = o(1), as
v
\ ⊂ C 6
x , then f . More generally, f O(1/xr) o(1/xr) implies v(f) =r.
v
→ ∞ ∈ C ∈ \
(iii) The functions in G (R,C) v are unbounded on every interval of the
∞ \F
form (n, ), n N.
∞ ∈
Part (i) shows that the v-infinitesimals are proper infinitesimals (with respect
to the order relation in ). Part (ii) shows that the finite but non-infinitesimal
M
functions (with respect to the order in ) are v-finite. Part (iii) shows that if f is
M
outside v and if f happens to be in order with all n N, then f is an infinitely
F | | ∈
large element of G (R,C). We should notice that v contains also infinitely large
∞ C
functions (along with finite and infinitesimal). For example, lnx is infinitely large
in the sense that n < lnx for all n N and 1/lnx is a positive infinitesimal in
∈
the sense that 0< 1/lnx < 1/n for all n N.
∈
Theorem 1 (Properties of v) The function v is a pseudovaluation on in
M
the sense that:
(i) v(f) = iff f .
∞ ∈ N
(ii) v(fg) v(f) + v(g). In particular, we have v(cg) = v(f) for all c
≥ ∈
C, c = 0 and also v(f/xr) = v(f)+r for all r R.
6 ∈
(iii) v(f g) min v(f),v(g) . Moreover, v(f)= v(g) implies
± ≥ { } 6
v(f g) = min v(f),v(g) .
± { }
In addition, v has the following properties:
(iv) v( f)= v(f) = v(f );
− | |
(v) v(1/f) = v(f) whenever f is invertible in ;
− M
(vi) f < g implies v(f) v(g).
| | | | ≥
Proof: The above properties follow directly from the properties of the “sup” in
the definition of v and we leave the verification to the reader. N
Theorem 2 (Rings and Ideals) (i) is convex subring (with zero divisors) of
M
G (R,C), where “convex” means that if f G (R,C) and g , then f g
∞ ∈ ∞ ∈ M | |≤ | |
implies f . Also is a convex ideal in .
∈ M N M
5
(ii) We have v = v v and v v = ∅. Besides, v is convex subring
F C ∪I C ∩I F
of and is a convex ideal in .
v v
M I F
(iii) The ring v contains a copy of the field of complex numbers C presented
F
by the constant-functions. Also we have C v 0 .
⊂ C ∪{ }
Proof: (i) follows immediately from the definitions of and .
M N
(ii) Suppose f,g , i.e. v(f) 0,v(g) 0. It follows v(f + g)
v
∈ F ≥ ≥ ≥
min v(f), v(g) 0 and v(fg) v(f)+v(g) 0, by Theorem 1, i.e. f +g, fg
{ } ≥ ≥ ≥ ∈
. Similarly it follows that is closed under the addition and multiplication.
v v
F I
Suppose also that h , i.e. v(h) > 0. We have v(fh) v(f)+v(h) > 0, i.e.
v
∈ I ≥
fh , as required.
v
∈ I
(iii) follows from the fact that v(c) = 0 for all c C, c= 0. N
∈ 6
Definition 2 (Pseudometric) We define a pseudometric dv : 2 R by
M →
d (f,g) = e v(f g) (with the understanding that e = 0). We denote by ( ,d )
v − − −∞ v
M
the corresponding pseudometric space.
Theorem 3 (Properties of the Pseudometric) For any f,g,h :
∈ M
(i) d (f,g) = 0 iff f g .
v
− ∈N
(ii) d (f,g) = d (g,f).
v v
(iii) d (f,g) max d (f,h),d (h,g) d (f,h)+d (h,g).
v v v v v
≤ { } ≤
(iv) , and coincide with the open unit ball, the closed unit ball and the
v v v
I F C
unit sphere in ( ,d ), respectively.
v
M
Proof: Adirectconsequenceofthepropertyofthepseudovaluationv(Theorem1).
We should mention that d (f,g) max d (f,h),d (h,g) is called ultrametric
v v v
≤ { }
inequality.
Theorem 4 (Convergence in ) Let (f ) be a sequence in .
n
M M
(i)Iff (inparticular, f = 0), thenlim f = f in( ,d ) iff lim v(f )=
n n v n n
∈ N →∞ M →∞
in R .
∞ ∪{∞}
(ii) If f / , then lim f = f in ( ,d ) implies that v(f ) = v(f) for
n n v n
∈ N →∞ M
all sufficiently large n.
Proof:
(i) limn fn = f in ( ,dv) iff limn d(fn,f)= 0 in R iff
limn e−v→(f∞n−f) = 0 in RMiff limn v(→fn∞ f)= in R iff
→∞ →∞ − ∞ ∪{∞}
lim v(f ) = , as required, since v(f f) = v(f ) for all sufficiently large
n n n n
→∞ ∞ −
n, by part (iii) of Theorem 1.
(ii) We have lim v(f f) = , by (i). Suppose (on the contrary) that
n n
→∞ − ∞
there exists a unbounded sequence (νn) in N such that v(fνn) 6= v(f) for all n. It
follows v(f f)= min v(f ),v(f) for all n, by part (iii) of Theorem 1, which
νn − { νn }
implies lim v(f ) = . Finally, it follows v(f) = , a contradiction. N
n→∞ νn ∞ ∞
6
Corollary 1 Let (rn) be a sequence in R such that limn rn = . Let
(f ) be a sequence in such that f = o(1∪/x{r∞n)}or f = O(1/xr→n)∞as x ∞ for
n n n
M → ∞
all n. Then lim f = 0 in ( ,d ).
n n v
→∞ M
Proof: The result follows, by part (i) of the above theorem, since in both cases
v(f ) r for all sufficiently large n. N
n n
≥
Example 3 Wehavelim 1 = 0in( ,d )sincev( 1 )= n asn .
n→∞ xn M v xn → ∞ → ∞
v
3 Linearly -Independent Sets
We should notice that the set of the v-constants is not closed under addition
v
C
and multiplication in . Ournext goal is to look for vector spaces inside 0 .
v
M C ∪{ }
If Φ , we denote by Span(Φ) the span of Φ over C (within ). Recall
⊂ M M
that Span(Φ) is the smallest vector subspace of containing Φ.
M
Definition 3 (v-Independence) We say that a set Φ is linearly v-
⊂ M
independent over C if Span(Φ) v 0 .
⊂ C ∪{ }
Let Φ be an linearly v-independent set over C. It is clear that Φ v 0 .
⊂ C ∪{ }
Also, every subset Ψ of Φ is also linearly v-independent over C.
Here is another (trivial) reformulation of the above definition:
Lemma 4 Let Φ . Then the following are equivalent:
⊂ M
(i) Φ is linearly v-independent over C.
(ii) v(f) = 0 for all f Span(Φ), f = 0.
∈ 6
(iii) There exists a vector subspace K of such that Φ K 0 .
v
M ⊆ ⊂ C ∪{ }
Example 4 The set of the complex numbers C is linearly v-independent (in a
trivial way). If f 0 , then the set f is also linearly v-independent over
v
∈ C ∪{ } { }
C. In particular, the set 0 is linearly v-independent over C.
{ }
Lemma 5 Eachofthefollowing setsislinearlyv-independent: Φ = sinx, cosx ,
1
{ }
Φ = lnx, ln(lnx), ln(ln(lnx)),... , Φ = lnx, ln2x, ln3x, ln4x,... ,
2 3
{ } { }
Φ = e iπx, e iπx lnx, e iπx ln2x, e iπx ln3x,... . The set Φ = Φ
4 ± ± ± ± 5 1
{ } ∪
Φ Φ Φ is also linearly v-independent.
2 3 4
∪ ∪
Proof: We shall prove that Φ is linearly v-independent. For the sake of conve-
2
nience we denote l (x) = lnx, l (x) = ln(lnx), l (x) = ln(ln(lnx)),.... So, we
1 2 3
have Φ = l (x) : n = 0,1,2,... . It suffices to prove that if n α l is a
non-zer2o line{anr combination, then v}( n α l )= 0. First note thatk=th1iskiskobvi-
k=1 k k P
ously true in the case n= 1, since each l (x) has valuation 0. If the sum has more
n
P
7
than one term, we just need to show that n α l has a “dominant term,” as
k=1 k k
x . (Technically, this is the greatest term in the polynomial with respect to
→ ∞ P
the lexicographic ordering on the sequence of its exponents.) For this it suffices
to show that m > k implies lim [(λ (x))p/(λ (x))q] = 0 whenever p > 0 and
x m k
→∞
q > 0, which is easily verified by l’Hopital’s Rule. The rest of the sets are treated
similarly and we leave the verification to the reader. N
Remark 1 (Linear Independence vs. Linear v-Independence) Weobserve
that the sets in the above four examples are also linearly independent over C
(in the usual sense). Notice that the set Φ = lnx, 2lnx is also linearly v-
{ }
independent but itis clearly linearly dependent. Conversely, the functions f(x)= 1
and g(x) = 1 + 1/x are linearly independent over C, but f and g are not v-
−
independent, since v(f + g) = v(1/x) = 1 (not = 0). We also observe that if
a set Φ is linearly v-independent over C, then the set Φ 0 is also linearly
∪ { }
v-independent over C (in sharp contrast to the case of linear independence).
4 Pseudostandard Part
We prove the existence of a particular type of linear homomorphism st from
v
F
into , with range in 0 , which is an extension of lim (considered also
v v x
F C ∪{ } →∞
as a linear homomorphism). The applications of this construction appbear in the
next section.
Definition 4 (Maximal Vector Spaces) Let be a vector subspace of . We
C M
say that is maximal in v if C v 0 and there is no a vector
C C ⊂ C ⊂ C ∪ { }
subspace of such that $ v 0 .
K M C K ⊂ C ∪{ }
Lemma 6 (Existence) Let Φ be a linearly v-independent subset of (Defini-
M
tion 3). Then there exists a vector subspace of which is maximal in and
v
C M C
which contains Φ.
Proof: Let (Φ) denote the set of all vector subspaces of such that C Φ
U K M ∪ ⊆
0 andlet (Φ) beordered by inclusion. Notice that (Φ) is non-empty
v
K ⊂ C ∪{ } U U
since Span(Φ) (Φ). We observe that every monotonic subset(chain) of (Φ)
∈U B U
is bounded from above by B. It follows that (Φ) has a maximal element
B U
(as desired), by Zorn’s lemm∈aB. N
C S
We shall sometimes refer to Span(Φ) as the “explicit” part of the space and
C
to Span(Φ) as the “implicit” part of .
C \ C
8
Theorem 5 (v-Completeness) Let be a vector subspace of which is max-
C M
imal in . Then:
v
C
(i) is v-complete in in the sense that: µ ( ) = , where µ ( ) is the
v v v v
C F C F C
v-monad of ((6), Section 2).
C
(ii) We have = in the sense that every f has a unique asymp-
v v v
F C⊕I ∈ F
totic expansion f = ϕ+dϕ, where ϕ and dϕ ((3), Section 2).
v
∈ C ∈ I
Proof: (i) We have µ ( ) , since v(ϕ+dϕ) min v(ϕ),v(dϕ) 0, by part
v v
C ⊆ F ≥ { } ≥
(iii) of Theorem 1. To show µ ( ) , suppose (on the contrary) that there
v v
C ⊇ F
exists g µ ( ), i.e. v(g) 0 and v(g ϕ) 0 for all ϕ . By letting
v v
∈ F \ C ≥ − ≤ ∈ C
ϕ =0, we conclude that v(g) = 0, i.e. g . It follows v(g ϕ) = 0 for all ϕ
v
∈C − ∈C
(by part (iii) of Theorem 1, again), i.e. g ϕ for all ϕ . Thus we have
v
− ∈ C ∈ C
$ Span( g ) v 0 , contradicting the maximality of .
C C ∪{ } ⊂ C ∪{ } C
(ii) The existence of the asymptotic expansion f = ϕ+ dϕ follows from (i).
To show the uniqueness, suppose that ϕ+dϕ = 0. It follows v(ϕ) = v( dϕ) =
−
v(dϕ) > 0 implying ϕ = dϕ = 0 since 0 . N
v
C ⊂ C ∪{ }
The above result justifies the following definition.
Definition 5 (Pseudostandard Part Mapping) Let be a vector subspace of
C
which is maximal in . We define st : , by st (ϕ + dϕ) = ϕ,
v v v
M C C F → F C
where ϕ , dϕ . We say that st is a pseudostandard part mapping
v
∈ C ∈ I C
determined by . We shall sometimes writbe simply st instead obf the more precise
C
st , suppressing thedependence on , wbhenthechoiceof isclearfromthe context.
C C C
b
Theorem 6 (Properties of st) Let be a vector subspace of which is max-
b
C M
imal in . Then:
v
C
(i) The pseudostandard parbt mapping st is a linear homomorphism from
v
C F
into with range ran(st ) = , i.e. st (αf +βg) = αst (f)+βst (g), for all
v
F C C C C C
f,g v and all α,β C. b
∈ F ∈
(ii) consists of thebfixed points of bst in , i.e. =b f bst (ϕ) = ϕ .
v v
C C F C { ∈ F | C }
Consequently, we have st st = st .
C ◦ C C
(iii) Suppose (in addition to the abbove) that Φ for someblinearly v-
⊂ C
independent subset Φ ofb (bDefinibtion 3). Then we have st (ϕ +dϕ) = ϕ for
M C
all ϕ Span(Φ) and all dϕ v. Consequently, the restriction st C v is
∈ ∈ I C| ⊕I
a (proper) standard part mapping (in the sense of nonstandabrd analysis – A. H.
Lightstone and A. Robinson [9]), i.e. st (c+dϕ) = c for all c C anbd all dϕ v
C ∈ ∈ I
(see the discussion after Lemma 3). Or, equivalently, st C v coincides with
C| ⊕I
limx in the sense that for every f b C v we have:
→∞ ∈ ⊕I
b
st (f)= lim f(x),
C x
→∞
b
9
Proof: (i) We have f = ϕ+dϕ and g =ψ+dψ forsome ϕ,ψ , dϕ, dψ , by
v
∈ C ∈ I
Theorem 5. Also αf+βg =αϕ+βψ+αdϕ+βdψ and it is clear that αϕ+βψ
∈C
(since is a vector space) and αdϕ +βdψ (since is an ideal in , by
v v v
C ∈ I I F
Theorem2). Thus st (αf+βg) = αϕ+βψ = αst(ϕ)+βst(ψ) = αst (f)+βst (g),
C C C
as required, since st (ϕ) = ϕ and st (ψ) = ψ, by the definition of st .
C C C
(ii) As we mentbioned already, st (ϕ) = ϕbfor all ϕb , by tbhe definitbion of
C ∈ C
st . Conversely, subppose that st (fb) = f for some f . It followsbf by the
v
C C ∈ F ∈ C
uniqueness of the asymptotic expanbsion f = ϕ+dϕ (part (ii) of Theorem 5).
b (iii) follows from the definitbion of st taking into account that C Φ .N
C ∪ ⊂ C
Example 5 (The Case Φ = Φ ) Let be a vector subspace of which is max-
5 bC M
imal in and which contains Φ , where Φ is the set defined in Lemma 5 (Sec-
v 5 5
C
tion 3). For the sake of simplicity we shall write simply st instead of the
more precise st . By the above theorem, we have st(ϕ+dϕ) = ϕ for all ϕ
C ∈ C
(hence, forall ϕ Span(Φ ))and alldϕ . Inparticular, forbeverydϕ and
5 v v
∈ ∈ I ∈I
every c C we hbave: st(c+dϕ) = c; st(c+1/lnx) =bc+1/lnx, st(sinx+dϕ) =
∈
sinx, st(cosx+dϕ) = cosx, st(sinxcosx+dϕ) = sinxcosx, st(lnx+dϕ) =
lnx, st(ln(lnx) + dϕb) = ln(lnx), bst(lnnx + dϕ) = lnnx, sbt(e iπxlnnx +
±
dϕ) = be iπxlnnx, and st (sinbx)e iπxlnnx+dϕ = (sinx)e ibπxlnnx, n =
± ± ±
0,1,2,.b... b b
(cid:0) (cid:1)
b
v
5 -Asymptotic Series
We consider a type of infinite series in which we call a v-asymptotic series and
M
illustrate this concept by examples.
Recall that a series ∞n=0fn is called asymptotic as x → ∞, if fn+1 = o(fn)
for all n. Notice that ifPan asymptotic series ∞n=0fn is in M (i.e. fn ∈ M for
all n), then we have v(f ) v(f ) for all n. In contrast, we have the following
n n+1
≤ P
similar concept:
Definition 6 (v-Asymptotic Series) LetF = ∞n=0fn beaseriesinM. Then:
(i) F is called a v-asymptotic series, as x , if it has the following two
→P∞
properties:
(a) The set xrnfn n 0 is linearly v-independent over C (Section 3),
{ | ≥ }
where r = v(f ) and we let (by convention) x 0= 0.
n n ∞
·
(b) The restriction of the sequence (r ) on the set n f = 0 is a strictly
n n
{ | 6 }
increasing (finite or infinite) sequence in R.
Every v-asymptotic series F can be written uniquely in its canonical form:
∞ ϕn(x)
(7) F = ,
xrn
n=0
X
10