Characterization of distributions having a value at a point in the sense of Robinson

0.12 MB·English

by  Hans Vernaeve

CHARACTERIZATION OF DISTRIBUTIONS HAVING A VALUE
AT A POINT IN THE SENSE OF ROBINSON
HANSVERNAEVEANDJASSONVINDAS
Abstract. WecharacterizeSchwartzdistributionshavingavalueatasingle
pointinthesenseintroducedbymeansofnonstandardanalysisbyA.Robin-
2
son. They appear to be distributions continuous in a neighborhood of the
1
point.
0
2

n
a
J
1. Introduction
3
1 In[9,§5.3],A.Robinsoninitiatedtheuseofnonstandardanalysisinthetheoryof

Schwartzdistributions. Amongotherthings,heintroducesnonstandardrepresenta-
]
A tives ofaSchwartzdistributionand, by meansof apropertyofthe representatives,
F he introduces a notion of point value of a distribution.
. A natural question to raise is how Robinson’s notion of point value is related to
h
the classical definition of point value in the sense of L ojasiewicz [5, 6]. Through
t
a investigation of this question, we arrived at a characterization which is the main
m
resultofthispaper: adistributionhasavalueatx ∈ΩinthesenseofRobinsoniff
0
[ it is a continuous function in a neighborhoodof x . Our characterizationimproves
0
an earlier result of P. Loeb [4] which has to assume the everywhere existence of
1
v Robinson point values (cf. section 3).
2 We mention that the concept of point value in the sense of L ojasiewicz has
5 shown to be of great importance in several areas of mathematical analysis and
9
its applications, such as spectral expansions [1, 11, 14], sampling theorems and
2
summability of cardinal series [12], edge detection from spectral data [2], or the
.
1 convergence of wavelet expansions [8, 13].
0
2
1
2. Notations
:
v
i By Ω, we always denote an open subset of Rn. We denote B(a,r) := {x∈ Rn :
X
|x−a|<r}.
r
a
2.1. Schwartzdistributions. WedenotebyD(Ω)thespaceoffunctionsinC∞(Ω)
withcompactsupportcontainedinΩ. ItsdualspaceD′(Ω)isthespaceofSchwartz
distributions on Ω. We denote the action of a linear map T: D(Ω)→C on an ele-
mentφ∈D(Ω)bymeansofthepairinghT,φi. Sometimesitisusefultodenotefunc-
tions anddistributions by means ofits actionona dummy variable(e.g., asin [1]);
wethendenotetheactionofT onφashT(x),φ(x)i. Thisallowsustowritechanges
of variables y = F(x) simply as hT(F(x)),φ(x)i = T(y),φ(F−1(y))· DF−1(y) .
We refer to [1, 10] for further information about Sc(cid:10)hwartz distribution(cid:12)s. (cid:12)(cid:11)
(cid:12) (cid:12)
2000 Mathematics Subject Classification. 26E35,46F10, 46S20.
Keywords and phrases. Schwartzdistributions,nonstandardanalysis,pointvalues.
J. Vindas gratefully acknowledges support by a Postdoctoral Fellowship of the Research
Foundation–Flanders (FWO,Belgium).
1
2 HANSVERNAEVEANDJASSONVINDAS
2.2. Nonstandard analysis. We work in the framework of nonstandard analysis
asintroducedbyRobinson[9]. We refer to[3]fora moreaccessibleintroductionto
nonstandardanalysis. As usual,iff: Rn →Rm is a function, wekeepthe notation
f forits canonicalextension∗f (andsimilarlyforrelationsonRn). Also∗ willbe
denoted by . We denote the set of all finite numbers in ∗C by Fin(∗C) aRnd write
x ≈ y if |x−R y| is infinitesimal (x,y ∈ ∗Rn). We write x / y if x ≤ y or x ≈ y
(x,y ∈∗R). We denote the standard part (a.k.a. shadow) by st.
3. Known results
Robinson works with real valued distributions on the real line, but the general-
ization to complex valued distributions onan open subset Ω of Rn is in most cases
straightforward. We say that a function f ∈ ∗C∞(Ω) represents (in the sense of
Robinson)a(notnecessarilycontinuous)linearmapT: D(Ω)→Cif fφ≈hT,φi
Ω
for each φ ∈ D(Ω). In fact, more general functions than ∗C∞(Ω)R-functions are
allowed as representatives, but ∗C∞(Ω)-functions suffice to develop distribution
theory by infinitesimal means. Robinson calls equivalence classes of functions rep-
resenting the same map T predistributions. We will identify the predistribution
with the map T.
Robinson calls a predistribution standard at x ∈ Ω if it has a representative f
0
that is S-continuous at x , i.e., such that f(x)≈f(x ) for each x≈x [9, p. 140].
0 0 0
He shows (with a slightly different proof):
Theorem 3.1 (Robinson). Let T be a linear map D(Ω)→C. If T has a represen-
tative f that is S-continuous at x ∈Ω, then f(x )∈Fin(∗C). Moreover, the value
0 0
stf(x ) does not depend on the chosen S-continuous representative.
0
Proof. Let ε∈R, ε >0. Since f is internal and S-continuous, we find by overspill
(see e.g. [3, 11.9.1]) on the set
{r∈∗R,r >0:(∀x∈∗Ω)|x−x |≤r =⇒ |f(x)−f(x )|≤ε}
0 0
thatthereexistsr ∈R, r >0suchthat|f(x)−f(x )|≤εforeachx∈∗B(x ,r)⊆
0 0
∗Ω. Now let φ ∈ D(B(x ,r)) with φ = 1. Let C := |φ| ∈ R. Then, since f
0 Ω Ω
represents T, R R
|hT,φi−f(x0)|≈(cid:12)(cid:12)(cid:12)(cid:12)Z∗Ωf(x)φ(x)dx−f(x0)(cid:12)(cid:12)(cid:12)(cid:12)=(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z∗B(x0,r)(f(x)−f(x0))φ(x)dx(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
≤ ε|φ(x)| dx=Cε.
Z
∗B(x0,r)
In particular, f(x ) ∈ Fin(∗C). For any g representing T and S-continuous at x ,
0 0
we have the same inequality (possibly only for some smaller r ∈ R, r > 0), so
|f(x )−g(x )|/2Cε. As ε is arbitrary, stf(x )=stg(x ). (cid:3)
0 0 0 0
The number stf(x ) is calledthe value (in the sense ofRobinson) ofthe predis-
0
tribution at x . P. Loeb [4] proves that if T admits a value g(x) at each x ∈ Ω,
0
then the resulting map g: Ω → C is continuous. In that case, ∗g represents T [9,
5.3.15], and hence T is a continuous function (as a regular Schwartz distribution).
Actually, Loeb’s result is a particular case of theorem 4.5, shown below.
A distribution T ∈ D′(Ω) admits the value c ∈ C at x ∈ Ω in the sense of
0
L ojasiewicz [5, 6] if lim T(x +εx) = c, where the limit is interpreted in the
ε→0 0
distributional sense, i.e., if
limhT(x +εx),φ(x)i =hc,φi=c φ, ∀φ∈D(Ω).
ε→0 0 ZΩ
CHARACTERIZATION OF DISTRIBUTIONS HAVING A VALUE AT A POINT IN THE SENSE OF ROBINSON3
ObservethatifT iscontinuousinaneighborhoodofx ,thenonereadilyverifies
0
that T has a L ojasiewicz value at x and its value is in fact T(x ) = c. On
0 0
the other hand, the converse is not true, in general, as shown by the function
T(x)=|x|−21 ei/x, which is unbounded at the origin but it admits the L ojasiewicz
value 0 at x = 0. More generally, any function of the form |x|−βei/|x|α, where
0
α,β > 0, uniquely determines a distribution that has L ojasiewicz value 0 at the
origin [5].
Of crucial importance for our result is the following theorem [6, §6.2]:
Theorem 3.2. Let T ∈D′(Ω) and x ∈Ω. If lim T(x +εx+λ) exists in
0 ε→0,λ→0 0
the distributional sense, i.e., lim hT(x +εx+λ),φ(x)i exists ∀φ ∈ D(Ω),
ε→0,λ→0 0
then T is a continuous function in a neighborhood of x .
0
4. New results
Lemma 4.1. Let T be a predistribution that is standard at x ∈ Ω. Then T is
0
a Schwartz distribution in an open neighborhood ω of x (i.e., T is a continuous
0 |ω
map D(ω)→C).
Proof. Let f be a representative of T that is S-continuous. As in the proof of
theorem 3.1, there exists r ∈ R, r > 0, such that |f(x)−f(x )| ≤ 1 for each
0
x∈∗B(x ,r)⊆∗Ω. Let φ∈D(B(x ,r)). Then
0 0
|hT,φi|≈(cid:12)(cid:12)(cid:12)Z∗B(x0,r)fφ(cid:12)(cid:12)(cid:12)≤Bs(xu0p,r)|f|µ(B(x0,r))Bs(xu0p,r)|φ|
(cid:12) (cid:12)
(cid:12) (cid:12) ≤ |f(x0)|+1 µ(B(x0,r)) sup |φ|,
(cid:0) (cid:1) B(x0,r)
where µ denotes the Lebesgue measure. Taking standard parts,
|hT,φi|≤ |stf(x )|+1 µ(B(x ,r)) sup |φ|.
0 0
(cid:0) (cid:1) B(x0,r)
Hence T is continuous. (cid:3)
|B(x0,r)
Definition. We call ψ ∈ ∗D(Rn) a strict nonstandard delta function (this name
corresponds to the standard notion of a strict delta net, see e.g. [7, §7]) if
(1) ψ =1
∗Rn
(2) Rψ(x)=0, ∀x6≈0
(3) |ψ|∈Fin(∗R).
∗Rn
R
Lemma 4.2. Let ψ be a strict nonstandard delta function. Let Ω be a (standard)
neighborhood of 0 and let f ∈∗C0(Ω) be S-continuous at 0. Then ψf ≈f(0).
∗Rn
R
Proof. As ψ is internal, there exists r ∈∗R, r ≈0 such that ψ(x)=0 if |x|≥r by
overspill. Then
f(0)− ψf = (f(0)−f(x))ψ(x)dx ≤sup|f(0)−f(x)| |ψ|≈0.
(cid:12) Z (cid:12) (cid:12)Z (cid:12) Z
(cid:12) ∗Rn (cid:12) (cid:12) ∗Rn (cid:12) x≤r ∗Rn
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:3)
In particular, each strict nonstandard delta function is a representative of the
delta distribution, since ∗φ is S-continuous for each φ∈D(Rn) (see e.g. [3, §7.1]).
Theorem 4.3. Let T be a predistribution that admits the value c (in the sense of
Robinson) at x ∈Ω. Then h∗T(x),ψ(x−x )i≈c for each strict delta function ψ.
0 0
4 HANSVERNAEVEANDJASSONVINDAS
Proof. Let ε ∈ R, ε > 0. In the proof of theorem 3.1, we showed that there exists
r ∈R, r >0 such that for each φ∈D(B(x ,r)) with φ=1 and |φ| =C, we
0 Ω Ω
have|hT,φi−c|≤(C+1)ε. By overspill,for eachφ∈R∗D(B(x ,r))Rwith φ=1
0 ∗Ω
and |φ|=C, we have |h∗T,φi−c|≤(C+1)ε. This holds in particulaRr for any
∗Ω
ε ∈ RR, ε > 0 and φ(x) = ψ(x−x ) with ψ a strict nonstandard delta function.
0
Hence h∗T(x),ψ(x−x )i≈c for such ψ. (cid:3)
0
From our main result, theorem 4.5, it will follow that the property in the state-
ment of the previous theorem actually characterizes predistributions admitting a
value at x in the sense of Robinson.
0
The previous theorem sheds a light on the relation between point values in the
sense of Robinson and point values in the sense of L ojasiewicz. By a nonstandard
characterization of limits (see e.g. [3, §7.3]) a distribution T ∈ D′(Ω) admits the
value c at x ∈Ω in the sense of L ojasiewicz iff h∗T(x),ψ(x−x )i≈c for each so-
0 0
called model delta function ψ, i.e., ψ(x) =ρ−nφ(x/ρ) with φ ∈ D(Rn), φ= 1,
Rn
ρ ∈ ∗R\{0}, ρ ≈ 0 (the name model delta function corresponds to theRstandard
notion of a model delta net, see e.g. [7, §7]).
Lemma 4.4. Let ω be an open subset of Ω and let x ∈ω. Then a linear map T:
0
D(Ω) → C admits the value c at x (in the sense of Robinson) iff T : D(ω) → C
0 |ω
admits the value c at x (in the sense of Robinson).
0
Proof. ⇒: immediate.
⇐: Letf ∈∗C∞(ω)bearepresentativeofT ,i.e., fφ≈hT,φi,∀φ∈D(ω),and
|ω ∗ω
suppose that f is S-continuous in x . Let g ∈ ∗C∞R(Ω) be any representative of T.
0
Thenf−g isarepresentativeofthe0-distributiononω. Letχ∈D(ω)withχ=1
|ω
on some (standard) neighborhood V of x . Then (f −g)χφ ≈ 0, ∀φ ∈ D(Ω),
0 ∗Ω
since χφ ∈ D(ω). So (f −g)χ is a representativeRof 0 on Ω, and fχ+g(1−χ)
is a representative of T on Ω which is equal to f in a (standard) neighborhood of
x . (cid:3)
0
Theorem 4.5. Let T be a predistribution and x ∈ Ω. Then T is standard at x
0 0
iff T is a continuous function in a neighborhood of x .
0
Proof. ⇒: by lemma 4.1, there exists an open neighborhood ω of x such that
0
T ∈ D′(ω). Let c denote the value of T at x (in the sense of Robinson). Let
0
φ∈D(ω) with φ=1. Let ε∈∗R, ε≈0 and λ∈∗Rn, λ≈0. Then, by theorem
ω
4.3, R
x−x −λ
∗T(x +εx+λ),φ(x) = ∗T(x),ε−nφ 0 ≈c,
0
(cid:10) (cid:11) D (cid:16) ε (cid:17)E
since ψ(x):=ε−nφ(x−λ)∈∗D(Rn) is a strict nonstandard delta function:
ε
1. by a change of variables, ψ = φ=1.
∗Rn ∗Rn
2. ψ(x)6=0 iff x−λ ∈suppφRiff x∈λR+εsuppφ; then in particular x≈0.
ε
3. |ψ|= |φ|∈R is finite.
∗Rn ∗Rn
AsRε and λ arRe arbitrary, by a nonstandard characterization of limits (see e.g. [3,
§7.3])
lim T(x +εx+λ),φ(x) =c
0
λ→0,ε→0(cid:10) (cid:11)
for each φ ∈ D(ω) with φ= 1. By theorem 3.2, T is a continuous function in a
ω
neighborhood of x . R
0
⇐: let ω be an open neighborhood of x and f ∈ C0(ω) such that T = f
0
on ω. Let g := f ⋆ ψ ∈ ∗C∞(ω′) for some neighborhood ω′ of x (⋆ denoting
0
convolution),whereψ isastrictnonstandarddeltafunction. Bylemma4.2,g(x)=
f(x−y)ψ(y)dy ≈ f(x) for each x∈∗ω′ (lemma 4.2 can be applied since f is
∗Rn
R
CHARACTERIZATION OF DISTRIBUTIONS HAVING A VALUE AT A POINT IN THE SENSE OF ROBINSON5
S-continuousat any x∈∗ω′ if the closureof ω′ is containedin ω, see e.g.[3, §7.1]).
In particular, g is S-continuous at x . Then for any φ∈D(ω′),
0
gφ−hT,φi = gφ− fφ ≤sup|g−f| |φ|≈0,
(cid:12)(cid:12)Z∗ω′ (cid:12)(cid:12) (cid:12)(cid:12)Z∗ω′ Z∗ω′ (cid:12)(cid:12) ∗ω′ Zω′
(cid:12) (cid:12) (cid:12) (cid:12)
so g repre(cid:12)sents T on ω′. B(cid:12)y le(cid:12)mma 4.4, T admi(cid:12)ts the value stg(x0) at x0. (cid:3)
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GhentUniversity,Krijgslaan 281,B-9000 Gent,Belgium.
E-mail address: [email protected]
GhentUniversity,Krijgslaan 281,B-9000 Gent,Belgium.
E-mail address: [email protected]