Bull. Soc. math. France,
117, 1989, p. 319–325.
SCHWARTZ’S THEOREM ON MEAN PERIODIC
VECTOR-VALUED FUNCTIONS
BY
François PARREAU and Yitzhak WEIT (*)
RÉSUMÉ. — Nous exposons une preuve plus simple du théorème de SCHWARTZ sur
les fonctions continues à valeurs dans CN .
ABSTRACT. — A simpler proof to SCHWARTZ’S theorem for CN -valued continuous
functions is provided.
1. Introduction and preliminaries
The theorem of L. SCHWARTZ on mean periodic functions of one
variable states that every closed translation-invariant subspace of the
space of continuous complex functions on R is spanned by the polynomialexponential functions it contains [4]. In [2, VII], J.-J. KELLEHER and B.-A.
TAYLOR provide a characterization of all closed submodes of CN -valued
entire functions of exponential type which have polynomial growth on R.
By duality, their result generalizes Schwartz’s Theorem to CN -valued
continuous functions.
Our goal is to provide a simple and a direct proof to this result.
C(R, CN ) denotes the space of continuous CN -valued functions on R,
with the topology of uniform convergence on compact sets. By a vectorvalued polynomial exponential in C(R, CN ), we mean a function of the
form eλx p(x), x ∈ R, where λ ∈ C and p is a polynomial in C(R, CN ).
Theorem. — Every translation-invariant closed subspace of C(R, CN )
is spanned by the vector-valued polynomial-exponential functions it contains.
For the theory of mean-periodic complex functions, we refer the reader
to [4], [1], [3]. We need the following notations and results.
(*) Texte reçu le 16 mai .
F. PARREAU, Dépt. de Mathématiques, C.S.P., Univ. Paris-Nord, 93430 Villetaneuse, France.
Y. WEIT, Dept. of Mathematics, Univ. of Haifa, Haifa 31999, Israël.
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F. PARREAU AND Y. WEIT
Let M0 (R) denote the space of complex Radon measures on R having
compact support. For µ ∈ M0 (R),
the Laplace transform µ̂ of µ is the
entire function defined by µ̂(z) = e−zx dµ(x), z ∈ C.
We remind that f ∈ C(R) is mean periodic if µ ∗ f = 0 for some
µ ∈ M0 (R), µ = 0. For f ∈ C(R), f − is the function defined by
f − (x) = f (x) if x ≤ 0 and f − (x) = 0 if x > 0. If f is mean-periodic,
µ ∈ M0 (R), µ = 0 and µ ∗ f = 0, then the function µ ∗ f − has compact
support and the meromorphic function
F = µ ∗ f − ˆ/µ̂,
which does not depend on the choice of µ, is defined to be the Laplace
transform of f ([3]).
The heart of our proof is the fact that F is entire only if f = 0 (see
[3, Theorem X]).
The dual of C(R, CN ) is the space M0 (R, CN ) of CN -valued Radon
measures on R having compact supports. One notices that M0 (R) is
an integral domain under the convolution product and M0 (R, CN ) is a
module over M0 (R) with the coordinatewise convolution. We denote the
duality by
N
µ, f =
µj ∗ fj (0)
j=1
for µ = (µj ) ∈ M0 (R, CN ) and f = (fj ) ∈ C(R, CN ). If f is a vectorvalued polynomial-exponential with
fj (x) =
m
()
αj x eλx
(1 ≤ j ≤ N ),
=0
we have
µ, f =
N m
j=1 =0
() ()
αj µ̂j (λ).
For any subset A of C(R, CN ) let
A⊥ = µ ∈ M0 (R, CN ) ; µ, f = 0
for all f ∈ A .
If V is a translation-invariant closed subspace of C(R, CN ), Sp(V ) denotes
the set of all vector-valued polynomial-exponentials that belong to V .
By duality, V is spanned by Sp(V ) if and only if Sp(V )⊥ ⊂ V ⊥ .
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Since V is translation-invariant, V ⊥ is a submodule of M0 (R, CN ) and
µ = (µj ) ∈ V ⊥ if and only if
N
µj ∗ fj = 0
for all f = (fj ) ∈ V.
j=1
2. Main result
In this section, V denotes a given translation-invariant closed subspace
of C(R, CN ). We have to prove µ, f = 0 for any µ ∈ Sp(V )⊥ and f ∈ V .
We need some more notation and three lemmas.
Let 0 ≤ r ≤ N be the rank of V ⊥ as a module over M0 (R). That means
r is the greatest integer for which there exists a system (σ )1≤≤r where
σ = (σ,j )1≤j≤N ∈ V ⊥ for 1 ≤ ≤ r and with a non-zero determinant of
order r. We shall suppose given such a system with, say,
ρ = det σ,j ; 1 ≤ , j ≤ r = 0.
One notices that ρ̂ is the non identically zero entire function given by
ρ̂(λ) = det ρ̂,j (λ) ; 1 ≤ , j ≤ r ,
λ ∈ C.
If r = 0, i.e. V ⊥ = {0}, we take for ρ the Dirac measure at 0 and ρ̂(λ) = 1,
λ ∈ C.
For µ = (µj ) ∈ M0 (R, CN ) let
µ1
σ1,1
∆j (µ) = det .
..
σr,1
and
. . . µr
. . . σ1,r
..
..
.
.
. . . σr,r
σ1,1
..
.
σ−1,1
τ (µ) = det µ1
σ+1,1
.
..
σr,1
µj σ1,j .. . σr,j
...
σ1,r .. ..
.
. . . . σ−1,r ...
µr . . . σ+1 , r .. ..
.
. ...
σr,r (for 1 ≤ j ≤ N )
(for 1 ≤ ≤ r).
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From the definition of r, for any µ ∈ V ⊥
(1)
(for 1 ≤ j ≤ N ).
∆j (µ) = 0
By expanding the ∆j (µ) along the last column, (1) is equivalent to
(2)
ρ ∗ µj =
r
τ (µ) ∗ σ,j
(for 1 ≤ j ≤ N ).
=1
Lemma 1. — Let λ ∈ C such that ρ̂(λ) = 0. For α = (αj ) ∈ CN , the
vector-exponential eλx · α belongs to V if and only if
N
(3)
αj σ̂,j (λ) = 0
1 ≤ ≤ r.
j=1
Proof. — Let α ∈ CN . We have eλx · α ∈ V if and only if, for every
µ = (µj ) ∈ V ⊥ ,
µ, eλx · α =
(4)
N
αj µ̂j (λ) = 0.
j=1
This proves the “only if” part. Conversly, since ρ̂(λ) = 0, (2) implies
that for any µ ∈ V ⊥ the equation in (4) is a linear combination of the
equations (3).
Lemma 2. — Let µ ∈ M0 (R, CN ). If µ, eλx · α = 0 for all λ ∈ C
such ρ̂(λ) = 0 and α ∈ CN such that eλx · α ∈ V , then ∆j (µ) = 0 for
1 ≤ j ≤ N.
Proof. — Let λ ∈ C with ρ̂(λ) = 0. If µ satisfies the hypothesis, the
solutions of (3) are solutions of (4), which implies that the determinants
∆j (µ)ˆ(λ) for 1 ≤ j ≤ N are equal to zero. Then, since ρ̂ and the ∆j (µ)ˆ
are entire functions and ρ̂ = 0, the ∆j (µ)ˆ are identically zero. Hence,
∆j (µ) = 0 for 1 ≤ j ≤ N .
Remark. — LEMMA 2 shows that any µ ∈ Sp(V )⊥ satisfies (1) and (2).
If r = 0, ∆j (µ) = µj for 1 ≤ j ≤ N ; hence Sp(V )⊥ = {0} if V ⊥ = {0}.
Lemma 3. — Let λ ∈ C, m ≥ 0 and µ ∈ Sp(V )⊥ . There exists ν ∈ V ⊥
such that
()
()
ν̂j (λ) = µ̂j (λ)
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SCHWARTZ’S THEOREM ON VECTOR-VALUED FUNCTIONS
323
()
Proof. — Suppose the element (µ̂j (λ))1<j<N, 0<−m of CN m does not
belong to the subspace
()
M (λ, m) = (ν̂j (λ))1≤j≤N, 0≤<m ; ν ∈ V ⊥ .
()
Then there exists (αj )1≤j≤N, 0≤<m such that
N m−1
j=1 =0
() ()
for ν ∈ V ⊥
αj ν̂j (λ) = 0
and
N m−1
() ()
αj µ̂j (λ) = 0.
j=1 =0
Then if
fj (x) =
m−1
()
αj x
(for 1 ≤ j ≤ N ),
=0
the polynomial-exponential f = (fj )1≤j≤N satisfies
(for ν ∈ V ⊥ ),
ν, f = 0
therefore f ∈ Sp(V ), and
µ, f = 0,
and we have a contradiction, since µ ∈ Sp(V )⊥ .
Proof of the THEOREM. — Let µ = (µj ) ∈ Sp(V )⊥ , f = (fj ) ∈ V and
g=
N
µj ∗ fj .
j=1
We have to prove that g = 0. By LEMMA 2, ∆j (µ) = 0 for 1 ≤ j ≤ N and
µ verifies (2) ; therefore
ρ∗
N
j=1
µj ∗ fj =
r
N
τ (µ) ∗
σ,j ∗ fj .
j=1
=1
For 1 ≤ ≤ r, since σ ∈ V ⊥ , we have
N
j=1
σ,j ∗ fj = 0. So
ρ ∗ g = 0.
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Hence g is mean-periodic and the Laplace transform G of g may be
defined by
G = ρ ∗ g − ˆ/ρ̂.
By ([3, Theorem X]) it is enough to prove that G is entire.
If [a, b]− is any interval that contains the supports of the µj (1 ≤ j ≤ N ),
µj ∗ fj (x) is equal to g(x) for x < a and 0 for x > b. Thus the function
s = g− −
N
j=1
µj ∗ fj−
has compact support. For 1 ≤ ≤ r, let
N
h =
σ,j ∗ fj− .
j=1
By the same argument, the functions h have compact supports and,
by (2),
ρ∗
So
N
j=1
µj ∗ fj− =
ρ ∗ g− =
G=
(5)
r
τ (µ) ∗ h .
=1
r
τ (µ) ∗ h + ρ ∗ s ;
=1
r
1
ρ̂
τ (µ)ˆ · ĥ + ŝ.
=1
The functions ŝ and ĥ (1 ≤ ≤ r) are entire, as Laplace transforms
of compactly supported functions.
νj ∗ fj = 0,
νj ∗ fj− has compact support,
For any ν ∈ V ⊥ , since
and it follows by (2) that the function
r
1
τ (ν)ˆ · ĥ
ρ̂
(6)
is entire.
=1
Let λ ∈ C and let m be the order of ρ̂ at λ. By LEMMA 3, we can choose
(k)
(k)
ν ∈ V ⊥ so that ν̂j (λ) = µ̂j (λ) for 1 ≤ j ≤ N , 0 ≤ k < m. Then the
functions (ν̂j − µ̂j )/ρ̂ for 1 ≤ j ≤ N and the functions
1
τ (ν)ˆ − τ (µ)ˆ
ρ̂
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SCHWARTZ’S THEOREM ON VECTOR-VALUED FUNCTIONS
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are analytic at λ. It follows from (5) and (6) that G is analytic at λ.
Since λ is arbitrary, G is entire. That completes the proof of the
Theorem.
BIBLIOGRAPHY
[1] KAHANE (J.-P.). — Sur les fonctions moyennes-périodiques bornées,
Ann. Inst. Fourier, t. 7, , p. 292–315.
[2] KELLHER (J.-J.) and TAYLOR (B.-A.). — Closed ideals in locally convex
algebras of analytic functions, J. Reine Angew. Math., t. 255, ,
p. 190–209.
[3] MEYER (Y.). — Algebraic Numbers and Harmonic Analysis (V-9). —
Amsterdam-London, North-Holland Publishing Co, .
[4] SCHWARTZ (L.). — Théorie générale des fonctions moyenne-périodiques, Ann. of Math., t. 48, , p. 857–928.
BULLETIN DE LA SOCIÉTÉ MATHÉMATIQUE DE FRANCE