SARAJEVO JOURNAL OF MATHEMATICS
Vol.13 (25), No.1, (2017), 61–70
DOI: 10.5644/SJM.13.1.04
AN APPLICATION OF MITTAG-LEFFLER LEMMA TO
THE L.F ALGEBRA OF C (∞) N-TEMPERED FUNCTIONS
ON R+
M. HEMDAOUI
Abstract. In this note, we show that a C (∞) N -tempered function f
on R+ can be extended as a C (∞) function fe in ] − ρ, +∞[+Ri (∀ρ > 0)
such that
∂
Dα fe(z)|{z=x} = 0, ∀α = (α1 , α2 ) ∈ N2 , ∀x ∈ R+ .
∂z
To get this result, we use Mittag-Leffler Lemma.
1. Introduction
Let f be a C (∞) function on R+ and N be a positive integer. f is N tempered if
sup δ0N (x)|f (n) (x)| < +∞, ∀n ∈ N,
x∈R+
1
.
1 + x2
(∞)
It is clair that the vector space denoted CN (R+ , δ0 , C) of N -tempered
functions on R+ equipped with the family of norms (k kN,n )n∈N defined by
where δ0 (x) = √
(∞)
f ∈ CN (R+ , δ0 , C) → kf kN,n = sup sup δ0N (x)|f (k) (x)|
0≤k≤n x∈R+
is a Frèchet space.
Let N , N 0 be two positive integers such that N ≤ N 0 , the natural injection
mapping
(∞)
(∞)
ΠN,N 0 : CN (R+ , δ0 , C) ,→ CN 0 (R+ , δ0 , C)
is obviously continuous.
2010 Mathematics Subject Classification. 46A04, 46A13, 46J05, 46J10,
Key words and phrases. Frèchet spaces - Frèchet algebras.
c 2017 by ANUBIH.
Copyright 62
M. HEMDAOUI
Consider the bilinear mapping
(∞)
(∞)
(∞)
(f, g) ∈ CN (R+ , δ0 , C) × CN 0 (R+ , δ0 , C) → f · g ∈ CN +N 0 (R+ , δ0 , C).
Applying Leibniz formula
D
n
n X
n
f (x)g(x)) =
f (n−p) (x) · g (p) (x).
p
p=0
So
0
δ0N +N (x)|Dn (f (x)g(x))| ≤ Cn sup sup δ0N (x)|f (k) (x)
x∈R+ 0≤k≤n
sup sup δ0N 0 (x)|g (p) (x).
x∈R+ 0≤p≤n
Then
kf · gkN +N 0 ≤ Cn kf kN · kgkN 0 ,
where
n X
n
n
Cn =
and
is the combination coefficient.
p
p
p=0
The inequality just above shows the continuity of the bilinear mapping
(∞)
(∞)
(∞)
(f, g) ∈ CN (R+ , δ0 , C) × CN 0 (R+ , δ0 , C) → f · g ∈ CN +N 0 (R+ , δ0 , C).
S
(∞)
(∞)
Consider the vector space CN (R+ , δ0 , C) = N ∈N CN (R+ , δ0 , C) of
C (∞) N-tempered functions on R+ equipped with the inductive limit topo(∞)
logy of Frèchet spaces CN (R+ , δ0 , C) is an L.F space. It is also a unital
commutative complex topological algebra since the bilinear mapping
[ (∞)
(f, g) ∈
CN (R+ , δ0 , C)
N ∈N
×
[
N ∈N
(∞)
CN (R+ , δ0 , C) → f · g ∈
[
(∞)
CN (R+ , δ0 , C)
N ∈N
is continuous.
First, for k ∈ N∗ we establish that a C (∞) function f on an nonempty
bounded open interval ]a, b[⊂ R can be extended as a function fek of class
C (k) in ]a, b[+Ri such that
∂
Dα fek (z)|{z=x} = 0, ∀α = (α1 , α2 ) ∈ N2 , /|α| = α1 + α2 ≤ k − 1, ∀x ∈]a, b[,
∂z
where
∂
1 ∂
∂
∂ |α|
= (
+ i ) and Dα =
, α = (α1 , α2 ) ∈ N2 .
∂z
2 ∂x
∂y
∂xα1 ∂y α2
AN APPLICATION OF MITTAG-LEFFLER LEMMA
63
At second, we use Mittag-Leffler lemma to get a C (∞) extension fe∞ of f
in ]a, b[+Ri such that
∂
Dα fe∞ (z)|{z=x} = 0, ∀α = (α1 , α2 ) ∈ N2 , ∀x ∈]a, b[.
(1.1)
∂z
S
(∞)
At last, we apply the result to the topological algebra N ∈N CN (R+ , δ0 , C).
B.Drost[2] used other technics to study extensions of ultra-differentiable
functions satisfying(1.1) on totally real compact submanifolds of open sets
in Cn .
S
(∞)
In a future note, we will show that the algebra N ∈N CN (R+ , δ0 , C)
equipped with its natural vector bornology is useful for functional calculus.
2. Notations and preliminary results
Let ]a, b[⊂ R be a nonempty bounded open interval and k ∈ N∗ . C (∞)
(]a, b[, C) denotes the vector space of C (∞) functions on ]a, b[ equipped with
its natural topology. It is a Frèchet space.
(k)
Qh∞ (]a, b[+Ri, C) denotes the vector space of functions f of class C (k)
(k ∈ N∗ ) in ]a, b[+Ri such that
(1) f |]a,b[ ∈ C (∞) (]a, b[, C).
∂
(2) Dα f (z)|{z=x} = 0, ∀α = (α1 , α2 ) ∈ N2 , /|α| = α1 + α2 ≤ k −
∂z
1, ∀x ∈]a, b[.
(k)
On Qh∞ (]a, b[+Ri, C), we consider the family of semi-norms defined by
f ∈ Qh(k)
∞ (]a, b[+Ri, C) → kf kn,K1 ,K2
= sup sup |f (p) (x)| + sup
0≤p≤n x∈K1
sup |Dα
|α|≤k−1 z∈K2
∂
f (z)|
∂z
where n ∈ N, K1 is a compact set in ]a, b[, K2 is a compact set in ]a, b[+Ri.
(k)
Qh∞ (]a, b[+Ri, C) equipped with the family of semi-norms k kn,K1 ,K2 is
a Frèchet space. The restriction mapping
(∞)
Rk : Qh(k)
(]a, b[, C)
∞ (]a, b[+Ri, C) → C
is obviously continuous.
We will show that the restriction Rk is surjective.
Lemma 2.1. Let ϕ be a C (∞) function with compact support in ] − 1, +1[
equal one on [− 21 , + 21 ] and (λn )n∈N be a sequence of strictly increasing positive real numbers such that limn→+∞ λn = +∞. Then, there exists a sequence of C (∞) functions (ϕn )n∈N satisfying
1
1
(∞)
(1) ϕn ∈ C
] − λn , + λn [ (∀n ∈ N) with compact support.
64
M. HEMDAOUI
(2) ϕn (x) = 1 if |x| ≤ 2λ1n , (∀n ∈ N).
(3) supp(ϕn+1 ) ⊂ supp(ϕn ), ∀n ∈ N.
Proof. For each positive integer n, consider the function defined by
ϕn (x) = ϕ(λn · x).
The sequence of functions (ϕn )n∈N satisfies Lemma 2.1.
Lemma 2.2. Let f ∈ C (∞) (]a, b[, C) with compact support K ⊂]a, b[ and
(k)
k ∈ N∗ . Then, f can be extended as a function fek in Qh∞ (]a, b[+Ri, C).
Proof. For each positive integer n ∈ N, put
Mn = sup |f (n) (x)|.
x∈K
Let (δn )n∈N be a sequence of positive real numbers satisfying
δn > sup(1, Mn ),
∀n ∈ N.
For n ∈ N and fix k ∈ N∗ . Consider the sequence
λn,k =
n+k
X
δi .
i=0
Then lim λn,k = +∞.
n→+∞
Let (ϕn,k )n∈N be a sequence of C (∞) functions satisfying Lemma 2.1 above
constructed with the sequence of positive real numbers (λn,k )n∈N above.
Consider the series
+∞
X
1 (n)
f (x)ϕn,k (y)(y · i)n .
n!
n=0
The series is locally finite. It converges at each point z = x + yi ∈]a, b[+Ri.
So it defines a function fek on ]a, b[+Ri.
Obviously, if y = 0, we have fek (x) = f (x), ∀x ∈]a, b[.
Since the series is locally finite, fek is a C (∞) function at each point z = x+yi
such that y 6= 0.
At each point z = x + yi such that y = 0, fek is of class C (k) . In fact:
Let α = (α1 , α2 ) ∈ N2 / |α| = α1 + α2 ≤ k − 1. The derivative of order α of
the general term of the series is
α
(n)
n
D f (x)ϕn,k (y)(yi)
= f (n+α1 ) (x)Dα2 ϕn,k (y)(yi)n .
AN APPLICATION OF MITTAG-LEFFLER LEMMA
65
By Leibniz formula we have
α2 X
α2
(α −p)
α2
n
ϕn,k2 (y) · n(n − 1) · · · (n − 1 − p)y n−p ,
D ϕn,k (y)y =
p
p=0
α2
where
is the combination coefficient.
p
So,
α
D 2 ϕn,k (y)y n k−1 n+1−k
X
(p)
1
k
−
1
≤ n!
sup
sup ϕ (λn,k · y)
.
p
λn,k
y∈[− 1 , 1 ] 0≤p≤k
p=0
λn,k λn,k
So
α
D 2 ϕn,k (y)y n ≤ n!Ck
where Ck = supy∈[−
1
, 1
λn,k λn,k
1
n+1−k
,
λn,k
P
(p)
ϕ (λn,k · y) k−1
sup
0≤p≤k
p=0
]
k−1
p
.
And so
n−k
1 α (n)
1 n+1−k
1
D f (x)ϕn,k (y)(yi)n ≤ Mn+α1 Ck
≤ Ck
.
n!
λn,k
δ0
P
1 α
(n) (x)ϕ
n is dominated by the converThe series +∞
n,k (y)(yi)
n=0 n! D f
k+1
P
1 n−k
gent geometrical series Ck +∞
= Ck (δδ00)−1 .
n=0 ( δ0 )
So, the series
+∞
X
1 α (n)
n
D f (x)ϕn,k (y)(y · i)
n!
n=0
converges uniformly in K + Ri, ∀α = (α1 , α2 ) ∈ N2 /|α| = α1 + α2 ≤ k.
Thus fek is of class C (k) in ]a, b[+Ri.
From the calculation
1 ∂
∂ e
∂ e
fk (x + yi) =
+i
fk (x + yi)
∂z
2 ∂x
∂y
+∞
+∞
1 X 1 (n+1)
i X 1 (n)
n
=
f
(x) ϕn,k (y)−ϕn+1,k (y) (iy) +
f (x)ϕ0n,k (y)(iy)n .
2
n!
2
n!
n=0
n=0
we get
(1) fek |]a,b[ = f.
∂
(2) Dα fek (z)|{z=x} = 0, ∀α = (α1 , α2 ) ∈ N2 , /|α| = α1 + α2 ≤ k −
∂z
1, ∀x ∈]a, b[.
66
M. HEMDAOUI
This ends the proof of Lemma 2.2.
Theorem 2.3. The restriction mapping
(∞)
Rk : Qh(k)
(]a, b[, C)
∞ (]a, b[+Ri, C) → C
is surjective.
Proof. Let (Uj )j∈J be a locally finite covering of the interval ]a, b[. Each Uj is
relatively compact in ]a, b[ and (φj )j∈J be a partition of unity subordinated
to the covering (Uj )j∈J .
Let f ∈ C (∞) (]a, b[, C) and j ∈ J. Put φj · f = fj .
fj is a C (∞) function in ]a, b[ with compact support. By lemma 2.2, fj can
(k)
be extended as a fek,j
X∈ Qh∞ (]a, b[+Ri, C).
(k)
The function fek =
fek,j ∈ Qh∞
(]a, b[+Ri, C).
j∈J
We want an extension fe∞ of f of class C (∞) in ]a, b[+Ri satisfying
∂ e
f∞ (z)|{z=x} = 0, ∀α = (α1 , α2 ) ∈ N2 , ∀x ∈]a, b[.
∂z
To get this result, we need the following Mittag-Leffler Lemma.
Dα
Lemma 2.4. For n ∈ N, let En00 , En , En0 be Frèchet spaces. in : En00 → En ;
sn : En → En0 be continuous linear mappings such that for each positive
integer n ∈ N the following sequence
0 → En00 → En → En0 → 0
is exact.
00
0
Let u00n : En+1
→ En00 ; un : En+1 → En ; u0n : En+1
→ En0 be continuous linear
mappings such that
un ◦ in+1 = in ◦ u00n ,
u0n ◦ sn+1 = sn ◦ un .
If each linear mapping u00n has a dense range, then the following sequence
0 → lim
En00 → lim
En → lim
En0 → 0
←
←
←
n
n
n
is exact.
We use Mittag-Leffler Lemma as follows:
(∞)
Let K be a nonempty compact set in ]a, b[, C0 (K, C) the Frèchet space
(k)
of C (∞) functions with support in K and Qh∞ (K + Ri, C) (k ∈ N∗ ) the
Frèchet space of functions f of class C (k) with support in K + Ri such that
(∞)
(1) f |K ∈ C0
(K, C).
AN APPLICATION OF MITTAG-LEFFLER LEMMA
67
∂
(2) Dα f (z)|{z=x} = 0, ∀x ∈ K, ∀α = (α1 , α2 ) ∈ N2 /|α| = α1 + α2 ≤
∂z
k − 1.
For each positive integer k ∈ N∗ , the restriction linear mapping
(∞)
Rk : Qh(k)
∞ (K + Ri, C) → C0
(K, C)
is obviously continuous.
(k)
The kernel Ker(Rk ) is a closed subspace of Qh∞ (K +Ri, C). It is a Frèchet
(k)
subspace of Qh∞ (K + Ri, C).
Consider the following commutative diagram of exact sequences.
O
O
u00
k−1
0
/ Ker(Rk )
O
uk−1
ik
/ Ker(Rk+1 )
O
Rk
/ C (∞) (K, C)
0
O
uk
k+1
/ Qh(k+1)
(K + Ri, C)
∞
O
/ C (∞) (K, C)
0
O
uk+1
.
/ ..
O
/0
j
R
ik+1
u00
k+1
0
j
/ Qh(k)
∞ (K + Ri, C)
O
u00
k
0
O
/0
j
.
/ ..
O
.
/ ..
O
/0
where
(k+1)
(k)
uk : Qh∞ (K + Ri, C) → Qh∞ (K + Ri, C) the continuous canonical injection.
(∞)
(∞)
j : C0 (K, C) → C0 (K, C) the continuous identity mapping.
00
uk : Ker(Rk+1 ) → Ker(Rk ) the continuous canonical injection.
(k)
ik : Ker(Rk ) → Qh∞ (K + Ri, C) the continuous canonical injection.
These continuous morphisms satisfy for every positive integer k ∈ N∗ .
ik ◦ u00k = uk ◦ ik+1
and Rk ◦ uk = j ◦ Rk+1 .
Each linear morphism u00k (k ∈ N) has obviously a dense range.
Using Mittag-Leffler lemma, we get the following exact sequence
(∞)
0 → lim
Ker(Rk ) → lim
Qh(k)
∞ (K + Ri, C) → C0
←
←
k
k
(K, C) → 0.
68
M. HEMDAOUI
(∞)
This means that each function f ∈ C0 (K, C) can be extended as a C (∞)
function fe∞ in K + Ri such that
∂
Dα fe∞ (z)|{z=x} = 0, ∀x ∈ K, ∀α = (α1 , α2 ) ∈ N2 .
∂z
(∞)
Hence, fe∞ ∈ Qh∞ (K + Ri, C).
Theorem 2.5. Let f ∈ C (∞) (]a, b[, C). It can be extended as a C (∞) function
fe∞ in ]a, b[+Ri such that:
∂ e
f∞ (z)|{z=x} = 0, ∀x ∈]a, b[, ∀α = (α1 , α2 ) ∈ N2 .
∂z
Proof. Let (Uj )j∈J be a locally finite covering of the open interval ]a, b[ and
(ϕj )j∈J be a partition of unity subordinated to the covering (Uj )j∈J .
Let f ∈ C (∞) (]a, b[, C), and j ∈ J. Put fj = ϕj · f.
fj is C (∞) function with compact support. By Mittag-Leffler Lemma there
(∞) in supp(ϕ · f ) + Ri such that
exists a function fg
∞,j of class C
j
(1) fe∞,j |supp(ϕj ·f ) = f.
∂
(2) Dα fe∞,j (z)|{z=x} = 0, ∀x ∈ supp(ϕj ), ∀α = (α1 , α2 ) ∈ N2 .
∂z
X
(∞)
The function fe∞ =
fg
∞,j ∈ Qh∞ (]a, b[+Ri, C) and satisfies
Dα
j∈J
(1) fe∞ |]a,b[ = f.
∂
(2) Dα fe∞ (z)|{z=x} = 0, ∀x ∈]a, b[, ∀α = (α1 , α2 ) ∈ N2 .
∂z
This ends the proof of theorem 2.5.
(∞)
3. Application to N-tempered function algebra CN (R+ , δ0 , C)
As seen in introduction
(∞)
CN (R+ , δ0 , C) =
(∞)
[
(∞)
CN (R+ , δ0 , C) = lim
C (R+ , δ0 , C).
→ N
N
N ∈N
It is the algebra of C (∞) N-tempered functions on R+ equipped with the
(∞)
inductive limit topology of Frèchet spaces CN (R+ , δ0 , C). An element f
in this algebra means that there exists a positive integer N such that f ∈
(∞)
CN (R+ , δ0 , C) and satisfies
∀n ∈ N,
sup δ0N (x)|f (n) (x)| < +∞.
x∈R+
The function f and all its derivative of any order have the same polynomially
growth at infinity.
AN APPLICATION OF MITTAG-LEFFLER LEMMA
69
(∞)
Theorem 3.1. Let f in CN (R+ , δ0 , C). There exists a function fe∞ of class
C (∞) in ] − ρ, +∞[+Ri (∀ρ > 0) such that:
(1) fe∞ |R+ = f.
∂
(2) Dα fe∞ (z)|{z=x} = 0, ∀x ∈ R+ , ∀α = (α1 , α2 ) ∈ N2 .
∂z
S
(∞)
(∞)
Proof. Let f ∈ N ∈N CN (R+ , δ0 , C) ⇒ ∃N ∈ N such that f ∈ CN (R+ ,
δ0 , C). By E. Borel theorem f can be extended as a C (∞) function f1 on R.
Let ϕ be a C (∞) function in R satisfying
1 if x ≥ 0
ϕ(x) =
0 if x ≤ −ρ (ρ > 0).
Multiply f1 by the function ϕ we get a function F of C (∞) with support in
] − ρ, +∞[.
Let ∈]0, 12 [ and consider the open interval ]1 − , 2 + [.
For k ∈ Z, consider the open interval 2k ]1 − , 2 + [.
The sequence of open intervals (2k ]1 − , 2 + [)k∈Z satisfy
T
n ]1 − , 2 + [ 2m ]1 − , 2 + [= ∅ if |m − n| ≥ 3.
(1) 2[
(2)
2n ]1 − , 2 + [=]0, +∞[.
n∈Z
Let φ be a positive function in C (∞) (R+ , C) with compact support in the
interval ]1 − , 2 + [ equal 1 on [1, 2].
For k ∈ Z and ρ ∈]0, +∞[, put
φk (x) = φ[2k (x + ρ)].
The function ϕ(x) =
of definition.
For k ∈ Z, put
P
k∈Z φk (x)
is positive and of class C (∞) on its domain
φk (x)
.
k0 ∈Z φk0 (x)
νk (x) = P
Then
X
νk (x) = 1,
∀x ∈] − ρ, +∞[.
k∈Z
(νn )n∈Z is a C (∞) partition of unity subordinated to the open covering
(2n ]1 − , 2 + [−ρ)n∈Z
of the open interval ] − ρ, +∞[.
Let n ∈ Z. Put f1,n = νn · F .
We get a C (∞) function with compact support in 2n ]1 − , 2 + [−ρ.
70
M. HEMDAOUI
So, by Theorem 2.5 above it can be extended as a C (∞) function fg
1,n in
(∞) n
Qh∞ 2 ]1 − , 2 + [−ρ + Ri .
P
The function fe∞ = n∈Z fg
1,n satisfies
(1) fe∞ |R+ = f .
∂
(2) Dα fe∞ (z)|{z=x} = 0, ∀x ∈ R+ , ∀α = (α1 , α2 ) ∈ N2 .
∂z
(∞)
Hence the function fe∞ ∈ Qh∞ ] − ρ, +∞[+Ri .
This ends the proof of Theorem 3.1.
Acknowledgements: The author would like to express his thanks to the
referee for his suggestions concerning the paper.
References
[1] J. Dieudonné, Elément d’analyse Moderne, 80 Tome 1, Gautier - Villard - Paris
1968.
[2] B. Drost, Holomorphic approximation of ultra differentiable functions, Math. Ann.,
257 (1981), 293–316.
[3] M. Hemdaoui, Calcul symbolique et l’opérateur de Laplace, Doctorat: Université
Libre de Bruxelles(ULB). Belgique. 03 Juin 1987.
[4] Hikosaburo Komatsu, Ultra ditributions I, structure theorems and caracterizations,
J. Fac. Sci. Univ. Tokyo Sec 1 A, 20 )1973), 25–105.
[5] J. Horvarth, Topological Vector Spaces and Distributions, Volume 1, Addison-Wesley
2985.
(Received: January 5, 2016)
(Revised: September 4, 2016)
Laboratoire Analyse Géométrie
et Applications(LAGA)
Université Mohamed Premier Faculté des Sciences
Département de Mathématiques & Informatique
Oujda
Maroc
[email protected]